A wave function or wavefunction (also more appropriately named a statefunction) in quantum mechanics describes the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of space and time. The Schrödinger equation describes how the wave function evolves over time. The wave function behaves qualitatively like other waves, like water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle duality.
The most common symbols for a wave function are ψ or Ψ (lower-case and capital psi).
Although ψ is a complex number, Template:Abs^{2} is real corresponding by Max Born's proposal to the probability density of finding a particle in a given place at a given time, if the particle's position is to be measured. Louis de Broglie in his later years proposed a real-valued wave function connected to the complex wave function by a proportionality constant and developed the de Broglie–Bohm theory.
The SI units for ψ depend on the system. For one particle in three dimensions, its units are m^{–3/2}. These unusual units are required so that an integral of Template:Abs^{2} over a region of three-dimensional space is a unitless probability (i.e., the probability that the particle is in that region). For different numbers of particles and/or dimensions, the units may be different, determined by dimensional analysis.^{[1]}
The wave function is central to quantum mechanics, because it is a fundamental postulate of quantum mechanics. It is the source of the mysterious consequences and philosophical difficulties in the interpretations of quantum mechanics—topics that continue to be debated even today.
Historical background
In the 1920s and 1930s, there were two divisions (so to speak) of theoretical physicists who simultaneously founded quantum mechanics: one for calculus and one for linear algebra. Those who used the techniques of calculus included Louis de Broglie, Erwin Schrödinger, Paul Dirac, Hermann Weyl, Oskar Klein, Walter Gordon, Douglas Hartree and Vladimir Fock. This hand of quantum mechanics became known as "wave mechanics". Those who applied the methods of linear algebra included Werner Heisenberg, Max Born, Wolfgang Pauli and John Slater. This other hand of quantum mechanics came to be called "matrix mechanics". Schrödinger subsequently showed that the two approaches were equivalent.^{[2]} In each case, the wavefunction was at the centre of attention in two forms, giving quantum mechanics its unity.
De Broglie could be considered the founder of the wave model in 1925, owing to his symmetric relation between momentum and wavelength: the De Broglie equation. Schrödinger searched for an equation that would describe these waves, and was the first to construct and publish an equation for which the wave function satisfied in 1926, based on classical energy conservation. Indeed it is now called the Schrödinger equation. However, no one, even Schrödinger and De Broglie, were clear on how to interpret it. What did this function mean?^{[3]}
Around 1924–27, Max Born, Heisenberg, Bohr and others provided the perspective of probability amplitude.^{[4]} This is the Copenhagen interpretation of quantum mechanics. There are many other interpretations of quantum mechanics, but this is considered the most important – since quantum calculations can be understood.
In 1927, Hartree and Fock made the first step in an attempt to solve the N-body wave function, and developed the self-consistency cycle: an iterative algorithm to approximate the solution. Now it is also known as the Hartree–Fock method.^{[5]} The Slater determinant and permanent (of a matrix) was part of the method, provided by Slater.
Interestingly, Schrödinger did encounter an equation for which the wave function satisfied relativistic energy conservation before he published the non-relativistic one, but it led to unacceptable consequences; negative probabilities and negative energies, so he discarded it.^{[6]}^{:3} In 1927, Klein, Gordon and Fock also found it, but taking a step further: incorporated the electromagnetic interaction into it and proved it was Lorentz-invariant. De Broglie also arrived at exactly the same equation in 1928. This wave equation is now known most commonly as the Klein–Gordon equation.^{[7]}
By 1928, Dirac deduced an equation from the first successful unification of special relativity and quantum mechanics, as applied to the electron – now called the Dirac equation. He found an unusual character of the wavefunction for this equation: it was not a single complex number, but a spinor.^{[5]} Spin automatically entered into the properties of the wavefunction. Although there were problems, Dirac was capable of resolving them. Around the same time Weyl also found his relativistic equation, which also had spinor solutions. Later other wave equations were developed: see Relativistic wave equations for further information.
Mathematical introduction
Wavefunctions as multi-variable functions – analytical calculus formalism
Multivariable calculus and analysis (study of functions, change etc.) can be used to represent the wavefunction in a number of situations. Superficially, this formalism is simple to understand for the following reasons.
- It is more directly intuitive to have probability amplitudes as functions of space and time. At every position and time coordinate, the probability amplitude has a value by direct calculation.
- Functions can easily describe wave-like motion, using periodic functions, and Fourier analysis can be readily done.
- Functions are easy to produce, visualize and interpret, due to the pictorial nature of the graph of a function (i.e. curves, Contour lines, and surfaces). When the situation is in a high number of dimensions (say 3-d space) – it is possible to analyze the function in a lower dimensional slice (say a 2-d plane) or contour plots of the function to determine the behaviour of the system within that confined region.
Although these functions are continuous, they are not deterministic; rather, they are probability distributions. Perhaps oddly, this approach is not the most general way to represent probability amplitudes. The more advanced techniques use linear algebra (the study of vectors, matrices, etc.) and, more generally still, abstract algebra (algebraic structures, generalizations of Euclidean spaces etc.).
Wave functions as an abstract vector space – linear/abstract algebra formalism
The set of all possible wave functions (at any given time) forms an abstract mathematical vector space. Specifically, the entire wave function is treated as a single abstract vector:
- $\backslash psi(\backslash mathbf\{r\})\; \backslash leftrightarrow\; |\backslash psi\backslash rangle$
where Template:Ket is a column vector written in bra–ket notation. The statement that "wave functions form an abstract vector space" simply means that it is possible to add together different wave functions, and multiply wave functions by complex numbers (see vector space for details). (Technically, because of the normalization condition, wave functions form a projective space rather than an ordinary vector space.) This vector space is infinite-dimensional, because there is no finite set of functions which can be added together in various combinations to create every possible function. Also, it is a Hilbert space, because the inner product of wave functions Ψ_{1}(x) and Ψ_{2}(x) can be defined as
- $\backslash langle\; \backslash Psi\_1\; |\; \backslash Psi\_2\; \backslash rangle\; \backslash equiv\; \backslash int\backslash limits\_\{-\backslash infty\}^\backslash infty\; d\; x\; \backslash ,\; \backslash Psi\_1^*(x)\backslash Psi\_2(x)\; ,$
where * denotes complex conjugate.
There are several advantages to understanding wave functions as elements of an abstract vector space:
- All the powerful tools of linear algebra can be used to manipulate and understand wave functions. For example:
- Linear algebra explains how a vector space can be given a basis, and then any vector can be expressed in this basis. This explains the relationship between a wave function in position space and a wave function in momentum space, and suggests that there are other possibilities too.
- Bra–ket notation can be used to manipulate wave functions.
- The idea that quantum states are vectors in a Hilbert space is completely general in all aspects of quantum mechanics and quantum field theory, whereas the idea that quantum states are complex-valued "wave" functions of space is only true in certain situations.
Introduction to vector formalism
Given an isolated physical system, the allowed states of this system (i.e. the states the system could occupy without violating the laws of physics) are part of a Hilbert space H. Some properties of such a space are
- If Template:Ket and Template:Ket are two allowed states, then aTemplate:Ket + bTemplate:Ket is also an allowed state, provided |a|^{2} + |b|^{2} = 1. (This condition is due to normalization, see below.)
- There is always an orthonormal basis of allowed states of the vector space H.
Physically, the nature of the inner product is dependent on the basis in use, because the basis is chosen to reflect the quantum state of the system.
When the basis is a countable set { Template:Ket } and orthonormal, that is
- $\backslash langle\; \backslash varepsilon\_i\; |\; \backslash varepsilon\_j\; \backslash rangle\; =\; \backslash delta\_\{ij\},$
then an arbitrary vector Template:Ket can be expressed as
- $|\; \backslash psi\; \backslash rangle\; =\; \backslash sum\_i\; c\_i\; |\; \backslash varepsilon\_i\; \backslash rangle,$
where the components are the (complex) numbers c_{i} = Template:Bra-ket This wave function is known as a discrete spectrum, since the bases are discrete.
When the basis is an uncountable set, the orthonormality condition holds similarly,
- $\backslash langle\; \backslash varepsilon\; |\; \backslash varepsilon\_0\; \backslash rangle\; =\; \backslash delta\; \backslash left\; (\; \backslash varepsilon\; -\; \backslash varepsilon\_0\; \backslash right\; ),$
then an arbitrary vector $|\; \backslash psi\; \backslash rangle$ can be expressed as
- $|\; \backslash psi\; \backslash rangle\; =\; \backslash int\; d\; \backslash varepsilon\; \backslash psi(\backslash varepsilon)\; |\; \backslash varepsilon\; \backslash rangle\; .$
where the components are the functions ψ(ε) = Template:Bra-ket This wave function is known as a continuous spectrum, since the bases are continuous.
Paramount to the analysis is the Kronecker delta, δ_{ij}, and the Dirac delta function, δ(ε − ε_{0}), since the bases used are orthonormal. More detailed discussion of wave functions as elements of vector spaces is below, following further definitions.
Requirements
The wavefunction must satisfy the following constraints for the calculations and physical interpretation to make sense:^{[8]}
- It must everywhere be finite.
- It must everywhere be a continuous function, and continuously differentiable (in the sense of distributions, for potentials that are not functions but are distributions, such as the dirac delta function).
- As a corollary, the function would be single-valued, else multiple probabilities occur at the same position and time, again unphysical.
- It must everywhere satisfy the relevant normalization condition, so that the particle/system of particles exists somewhere with 100% certainty.
If these requirements are not met, it's not possible to interpret the wavefunction as a probability amplitude; the values of the wavefunction and its first order derivatives may not be finite and definite (with exactly one value), i.e. probabilities can be infinite and multiple-valued at any one position and time – which is nonsense, as it does not satisfy the probability axioms. Furthermore, when using the wavefunction to calculate a measurable observable of the quantum system without meeting these requirements, there will not be finite or definite values to calculate from – in this case the observable can take a number of values and can be infinite. This is unphysical and not observed when measuring in an experiment. Hence a wavefunction is meaningful only if these conditions are satisfied.
Information about quantum systems
Although the wavefunction contains information, it is a complex number valued quantity; only its relative phase and relative magnitude can be measured. It does not directly tell anything about the magnitudes or directions of measurable observables. An operator extracts this information by acting on the wavefunction ψ. For details and examples on how quantum mechanical operators act on the wave function, commutation of operators, and expectation values of operators; see Operator (physics).
Definition (single spin-0 particle in one spatial dimension)
Travelling waves of a free particle.
The
real parts of position and momentum wave functions
Ψ(x) and
Φ(p), and corresponding probability densities
|Ψ(x)|^{2} and
|Φ(p)|^{2}, for one spin-0 particle in one
x or
p dimension. The wavefunctions shown are continuous, finite, single-valued and normalized. The colour opacity (%) of the particles corresponds to the probability density (
not the wavefunction) of finding the particle at position
x or momentum
p.
Position-space wavefunction
For now, consider the simple case of a single particle, without spin, in one spatial dimension. (More general cases are discussed below). The state of such a particle is completely described by its wave function:
- $\backslash Psi(x,t)$,
where x is position and t is time. This function is complex-valued, meaning that Ψ(x, t) is a complex number.
If the particle's position is measured, its location is not deterministic, but is described by a probability distribution. The probability that its position x will be in the interval [a, b] (meaning a ≤ x ≤ b) is:
- $P\_\{a\backslash le\; x\backslash le\; b\}\; =\; \backslash int\backslash limits\_a^b\; d\; x\backslash ,|\backslash Psi(x,t)|^2$
where t is the time at which the particle was measured. In other words, |Ψ(x, t)|^{2} is the probability density that the particle is at x, rather than some other location.
This leads to the normalization condition:
- $\backslash int\backslash limits\_\{-\backslash infty\}^\backslash infty\; d\; x\; \backslash ,\; |\backslash Psi(x,t)|^2\; =\; 1$,
because if the particle is measured, there is 100% probability that it will be somewhere.
Momentum-space wavefunction
Main article:
Momentum space
The particle also has a wave function in momentum space:
- $\backslash Phi(p,t)$
where p is the momentum in one dimension, which can be any value from −∞ to +∞, and t is time. If the particle's momentum is measured, the result is not deterministic, but is described by a probability distribution:
- $P\_\{a\backslash le\; p\backslash le\; b\}\; =\; \backslash int\backslash limits\_a^b\; d\; p\; \backslash ,\; |\backslash Phi(p,t)|^2$,
analogous to the position case.
The normalization condition is also similar:
- $\backslash int\backslash limits\_\{-\backslash infty\}^\{\backslash infty\}\; d\; p\; \backslash ,\; \backslash left\; |\; \backslash Phi\; \backslash left\; (\; p,\; t\; \backslash right\; )\; \backslash right\; |^2\; =\; 1.$
Relation between wavefunctions
The position-space and momentum-space wave functions are Fourier transforms of each other, therefore both contain the same information, and either one alone is sufficient to calculate any property of the particle. For one dimension:^{[9]}
- $\backslash begin\{align\}\; \backslash Phi(p,t)\; \&\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\backslash hbar\}\}\backslash int\backslash limits\_\{-\backslash infty\}^\backslash infty\; d\; x\; \backslash ,\; e^\{-ipx/\backslash hbar\}\; \backslash Psi(x,t)\backslash \backslash $
&\upharpoonleft \downharpoonright\\
\Psi(x,t) & = \frac{1}{\sqrt{2\pi\hbar}}\int\limits_{-\infty}^\infty d p \, e^{ipx/\hbar} \Phi(p,t).
\end{align}
Sometimes the wave-vector k is used in place of momentum p, since they are related by the de Broglie relation
- $p\; =\; \backslash hbar\; k,$
and the equivalent space is referred to as k-space. Again it makes no difference which is used since p and k are equivalent – up to a constant. In practice, the position-space wavefunction is used much more often than the momentum-space wavefunction.
Example of normalization
A particle is restricted to a 1D region between x = 0 and x = L; its wave function is:
- $\backslash begin\{align\}$
\Psi (x,t) & = Ae^{i(kx-\omega t)}, & 0 \leq x \leq L \\
\Psi (x,t) & = 0, & x < 0, x > L \\
\end{align} .
To normalize the wave function we need to find the value of the arbitrary constant A; solved from
- $\backslash int\backslash limits\_\{-\backslash infty\}^\{\backslash infty\}\; dx\; \backslash ,\; |\backslash Psi|^2\; =\; 1\; .$
From Ψ, we have |Ψ|^{2};
- $|\; \backslash Psi\; |\; ^2\; =\; A^2\; e^\{i(kx\; -\; \backslash omega\; t)\}\; e^\{-i(kx\; -\; \backslash omega\; t)\}\; =A^2\; ,$
so the integral becomes;
- $\backslash int\backslash limits\_\{-\backslash infty\}^0\; dx\; \backslash cdot\; 0\; +\; \backslash int\backslash limits\_0^L\; dx\; \backslash ,\; A^2\; +\; \backslash int\backslash limits\_L^\backslash infty\; dx\; \backslash cdot\; 0\; =\; 1\; ,$
therefore the constant is;
- $A^2\; L\; =\; 1\; \backslash rightarrow\; A\; =\; \backslash frac\{1\}\{\backslash sqrt\{L\}\}\; .$
The normalized wave function (in the region) is then given by;
- $\backslash Psi\; (x,t)\; =\; \backslash frac\{1\}\{\backslash sqrt\{L\}\}\; e^\{i(kx-\backslash omega\; t)\},\; \backslash quad\; 0\; \backslash leq\; x\; \backslash leq\; L.$
Definition (other cases)
Many spin-0 particles in one spatial dimension
The previous wavefunction can be generalized to incorporate N particles in one dimension:
- $\backslash Psi(x\_1,x\_2,\backslash cdots\; x\_N,\; t)$,
The probability that particle 1 is in an x-interval R_{1} = [a_{1},b_{1}] and particle 2 in interval R_{2} = [a_{2},b_{2}] etc., up to particle N in interval R_{N} = [a_{N},b_{N}], all measured simultaneously at time t, is given by:
- $P\_\{x\_1\backslash in\; R\_1,x\_2\backslash in\; R\_2\; \backslash cdots\; x\_N\backslash in\; R\_N\}\; =\; \backslash int\backslash limits\_\{a\_1\}^\{b\_1\}\; d\; x\_1\; \backslash int\backslash limits\_\{a\_2\}^\{b\_2\}\; d\; x\_2\; \backslash cdots\; \backslash int\backslash limits\_\{a\_N\}^\{b\_N\}\; d\; x\_N\; |\; \backslash Psi(x\_1\; \backslash cdots\; x\_N,t)|^2$
The normalization condition becomes:
- $\backslash int\backslash limits\_\{-\backslash infty\}^\backslash infty\; d\; x\_1\; \backslash int\backslash limits\_\{-\backslash infty\}^\backslash infty\; d\; x\_2\; \backslash cdots\; \backslash int\backslash limits\_\{-\backslash infty\}^\backslash infty\; d\; x\_N\; |\backslash Psi(x\_1\; \backslash cdots\; x\_N,t)|^2\; =\; 1$.
In each case, there are N one-dimensional integrals, one for each particle.
One spin-0 particle in three spatial dimensions
Position space wavefunction
The position-space wave function of a single particle in three spatial dimensions is similar to the case of one spatial dimension above:
- $\backslash Psi(\backslash mathbf\{r\},t)$
where r is the position in three-dimensional space (r is short for (x, y, z)), and t is time. As always Ψ(r, t) is a complex number. If the particle's position is measured at time t, the probability that it is in a region R is:
- $P\_\{\backslash mathbf\{r\}\backslash in\; R\}\; =\; \backslash int\backslash limits\_R\; d^3\backslash mathbf\{r\}\; \backslash ,\; \backslash left\; |\backslash Psi(\backslash mathbf\{r\},t)\; \backslash right\; |^2$
(a three-dimensional integral over the region R, with differential volume element d^{3}r, also written "dV" or "dx dy dz"). The normalization condition is:
- $\backslash int\backslash limits\_$Template:\rm all \, space \left | \Psi(\mathbf{r},t)\right |^2 d ^3\mathbf{r} = 1,
where the integrals are taken over all of three-dimensional space.
Momentum space wavefunction
There is a corresponding momentum space wavefunction for three dimensions also:
- $\backslash Phi(\backslash mathbf\{p\},t)$
where p is the momentum in 3-dimensional space, and t is time. This time there are three components of momentum which can have values −∞ to +∞ in each direction, in Cartesian coordinates (x, y, z).
The probability of measuring the momentum components p_{x} between a and b, p_{y} between c and d, and p_{z} between e and f, is given by:
- $P\_\{p\_x\backslash in[a,b],p\_y\backslash in[c,d],p\_z\backslash in[e,f]\}\; =\; \backslash int\backslash limits\_e^f\; dp\_z\backslash ,\backslash int\backslash limits\_c^d\; d\; p\_y\; \backslash ,\; \backslash int\backslash limits\_a^b\; d\; p\_x\; \backslash ,\; \backslash left\; |\; \backslash Phi\; \backslash left\; (\; \backslash mathbf\{p\},\; t\; \backslash right\; )\; \backslash right\; |^2\; ,$
hence the normalization:
- $\backslash int\backslash limits\_$Template:\rm all \, space d^3\mathbf{p} \, \left | \Phi \left ( \mathbf{p}, t \right ) \right |^2 = 1.
analogous to space, d^{3}p = dp_{x}dp_{y}dp_{z} is a differential 3-momentum volume element in momentum space.
Relation between wavefunctions
The generalization of the previous Fourier transform is^{[10]}
- $\backslash begin\{align\}\; \backslash Phi(\backslash mathbf\{p\},t)\; \&\; =\; \backslash frac\{1\}\{\backslash sqrt\{\backslash left(2\backslash pi\backslash hbar\backslash right)^3\}\}\backslash int\backslash limits\_$Template:\rm all \, space d ^3\mathbf{r} \, e^{-i \mathbf{r}\cdot \mathbf{p} /\hbar} \Psi(\mathbf{r},t) \\
&\upharpoonleft \downharpoonright\\
\Psi(\mathbf{r},t) & = \frac{1}{\sqrt{\left(2\pi\hbar\right)^3}}\int\limits_Template:\rm all \, space d^3\mathbf{p} \, e^{i \mathbf{r}\cdot \mathbf{p} /\hbar} \Phi(\mathbf{p},t) .
\end{align}
Many spin-0 particles in three spatial dimensions
When there are many particles, in general there is only one wave function, not a separate wave function for each particle. The fact that one wave function describes many particles is what makes quantum entanglement and the EPR paradox possible. The position-space wave function for N particles is written:^{[5]}
- $\backslash Psi(\backslash mathbf\{r\}\_1,\backslash mathbf\{r\}\_2\; \backslash cdots\; \backslash mathbf\{r\}\_N,t)$
where r_{i} is the position of the ith particle in three-dimensional space, and t is time. If the particles' positions are all measured simultaneously at time t, the probability that particle 1 is in region R_{1} and particle 2 is in region R_{2} and so on is:
- $P\_\{\backslash mathbf\{r\}\_1\backslash in\; R\_1,\backslash mathbf\{r\}\_2\backslash in\; R\_2\; \backslash cdots\; \backslash mathbf\{r\}\_N\backslash in\; R\_N\}\; =\; \backslash int\backslash limits\_\{R\_1\}\; d\; ^3\backslash mathbf\{r\}\_1\; \backslash int\backslash limits\_\{R\_2\}\; d\; ^3\backslash mathbf\{r\}\_2\backslash cdots\; \backslash int\backslash limits\_\{R\_N\}\; d\; ^3\backslash mathbf\{r\}\_N\; |\backslash Psi(\backslash mathbf\{r\}\_1\; \backslash cdots\; \backslash mathbf\{r\}\_N,t)|^2$
The normalization condition is:
- $\backslash int\backslash limits\_$Template:\rm all \, space d ^3\mathbf{r}_1 \int\limits_Template:\rm all \, space d ^3\mathbf{r}_2\cdots \int\limits_Template:\rm all \, space d ^3\mathbf{r}_N |\Psi(\mathbf{r}_1 \cdots \mathbf{r}_N,t)|^2 = 1
(altogether, this is 3N one-dimensional integrals).
For N interacting particles, i.e. particles which interact mutually and constitute a many-body system, the wavefunction is a function of all positions of the particles and time, it can't be separated into the separate wavefunctions of the particles. However, for non-interacting particles, i.e. particles which do not interact mutually and move independently, in a time-independent potential, the wavefunction can be separated into the product of separate wavefunctions for each particle:^{[8]}
- $\backslash Psi\; =\; \backslash phi(t)\backslash prod\_\{i=1\}^N\backslash psi(\backslash mathbf\{r\}\_i)\; =\; \backslash phi(t)\backslash psi(\backslash mathbf\{r\}\_1)\backslash psi(\backslash mathbf\{r\}\_2)\backslash cdots\backslash psi(\backslash mathbf\{r\}\_N).$
One particle with spin in three dimensions
For a particle with spin, the wave function can be written in "position-spin-space" as:
- $\backslash Psi(\backslash mathbf\{r\},s\_z,t)$
where r is a position in three-dimensional space, t is time, and s_{z} is the spin projection quantum number along the z axis. (The z axis is an arbitrary choice; other axes can be used instead if the wave function is transformed appropriately, see below.) The s_{z} parameter, unlike r and t, is a discrete variable. For example, for a spin-1/2 particle, s_{z} can only be +1/2 or −1/2, and not any other value. (In general, for spin s, s_{z} can be s, s – 1,...,–s.) If the particle's position and spin is measured simultaneously at time t, the probability that its position is in R_{1} and its spin projection quantum number is a certain value s_{z} = m is:
- $P\_\{\backslash mathbf\{r\}\backslash in\; R,s\_z=m\}\; =\; \backslash int\backslash limits\_\{R\}\; d\; ^3\backslash mathbf\{r\}\; |\backslash Psi(\backslash mathbf\{r\},t,m)|^2$
The normalization condition is:
- $\backslash sum\_\{\backslash mathrm\{all\backslash ,\; \}s\_z\}\; \backslash int\backslash limits\_$Template:\rm all \, space |\Psi(\mathbf{r},t,s_z)|^2 d ^3\mathbf{r} = 1.
Since the spin quantum number has discrete values, it must be written as a sum rather than an integral, taken over all possible values.
Many particles with spin in three dimensions
Likewise, the wavefunction for N particles each with spin is:
- $\backslash Psi(\backslash mathbf\{r\}\_1,\; \backslash mathbf\{r\}\_2\; \backslash cdots\; \backslash mathbf\{r\}\_N,\; s\_\{z\backslash ,1\},\; s\_\{z\backslash ,2\}\; \backslash cdots\; s\_\{z\backslash ,N\},\; t)$
The probability that particle 1 is in region R_{1} with spin s_{z1} = m_{1} and particle 2 is in region R_{2} with spin s_{z2} = m_{2} etc. reads (probability subscripts now removed due to their great length):
- $P\; =\; \backslash int\backslash limits\_\{R\_1\}\; d\; ^3\backslash mathbf\{r\}\_1\; \backslash int\backslash limits\_\{R\_2\}\; d\; ^3\backslash mathbf\{r\}\_2\backslash cdots\; \backslash int\backslash limits\_\{R\_N\}\; d\; ^3\backslash mathbf\{r\}\_N\; \backslash left\; |\; \backslash Psi\backslash left\; (\backslash mathbf\{r\}\_1\; \backslash cdots\; \backslash mathbf\{r\}\_N,m\_1\backslash cdots\; m\_N,t\; \backslash right\; )\; \backslash right\; |^2$
The normalization condition is:
- $\backslash sum\_\{s\_\{z\backslash ,N\}\}\; \backslash cdots\; \backslash sum\_\{s\_\{z\backslash ,2\}\}\; \backslash sum\_\{s\_\{z\backslash ,1\}\}\; \backslash int\backslash limits\_$Template:\rm all \, space d ^3\mathbf{r}_1 \int\limits_Template:\rm all \, space d ^3\mathbf{r}_2\cdots \int\limits_Template:\rm all \, space d ^3 \mathbf{r}_N \left | \Psi \left (\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) \right |^2 = 1
Now there are 3N one-dimensional integrals followed by N sums.
Again, for non-interacting particles in a time-independent potential the wavefunction is the product of separate wavefunctions for each particle:^{[8]}
- $\backslash Psi\; =\; \backslash phi(t)\backslash prod\_\{i=1\}^N\backslash psi(\backslash mathbf\{r\}\_i,s\_\{z\backslash ,i\})\; =\; \backslash phi(t)\backslash psi(\backslash mathbf\{r\}\_1,s\_\{z\backslash ,1\})\backslash psi(\backslash mathbf\{r\}\_2,s\_\{z\backslash ,2\})\backslash cdots\backslash psi(\backslash mathbf\{r\}\_N,s\_\{z\backslash ,N\}).$
Wavefunction symmetry and antisymmetry
In quantum mechanics there is a fundamental distinction between identical particles and distinguishable particles. For example, any two electrons are fundamentally indistinguishable from each other; the laws of physics make it impossible to "stamp an identification number" on a certain electron to keep track of it.^{[11]} This translates to a requirement on the wavefunction: For example, if particles 1 and 2 are indistinguishable, then:
- $\backslash Psi\; \backslash left\; (\; \backslash mathbf\{r\},\backslash mathbf\{r\text{'}\},\backslash mathbf\{r\}\_3,\backslash mathbf\{r\}\_4,\backslash cdots\; \backslash right\; )\; =\; \backslash left\; (\; -1\; \backslash right\; )^\{2s\}\; \backslash Psi\; \backslash left\; (\; \backslash mathbf\{r\text{'}\},\backslash mathbf\{r\},\backslash mathbf\{r\}\_3,\backslash mathbf\{r\}\_4,\backslash cdots\; \backslash right\; )$
where s is the spin quantum number of the particle: integer for bosons (s = 1, 2, 3...) and half-integer for fermions (s = 1/2, 3/2...).
The wavefunction is said to be symmetric (no sign change) under boson interchange and antisymmetric (sign changes) under fermion interchange. This feature of the wavefunction is known as the Pauli principle.
Normalization invariance
It is important that the properties associated with the wave function are invariant under normalization. If normalization of a wave function changed the properties, the process becomes pointless as we still cannot yield any information about the particle associated with the non-normalized wave function.
All properties of the particle, such as momentum, energy, expectation value of position, associated probability distributions etc., are solved from the Schrödinger equation (or other relativistic wave equations). The Schrödinger equation is a linear differential equation, so if Ψ is normalized and becomes AΨ (A is the normalization constant), then the equation reads:
- $\backslash hat\{H\}\; (A\backslash Psi)\; =\; i\backslash hbar\backslash frac\{\backslash partial\; \}\{\backslash partial\; t\}(A\backslash Psi)\; \backslash rightarrow\; \backslash hat\{H\}\; \backslash Psi\; =\; i\backslash hbar\backslash frac\{\backslash partial\; \}\{\backslash partial\; t\}\backslash Psi$
which is the original Schrödinger equation. That is to say, the Schrödinger equation is invariant under normalization, and consequently associated properties are unchanged.
Wavefunctions as vector spaces
Discrete components A_{k} of a complex vector Template:Ket = ∑_{k} A_{k}Template:Ket, which belongs to a countably infinite-dimensional Hilbert space; there are countably infinitely many k values and basis vectors Template:Ket.
Continuous components
ψ(x) of a complex vector
Template:Ket = ∫dx ψ(x)Template:Ket, which belongs to an
uncountably infinite-dimensional
Hilbert space; there are uncountably infinitely many
x values and basis vectors
Template:Ket.
Components of complex vectors plotted against index number; discrete k and continuous x. Two probability amplitudes out of infinitely many are highlighted.
Main article: Quantum state
As explained above, quantum states are always vectors in an abstract vector space (technically, a complex projective Hilbert space). For the wave functions above, the Hilbert space usually has not only infinite dimensions, but uncountably infinitely many dimensions. However, linear algebra is much simpler for finite-dimensional vector spaces. Therefore it is helpful to look at an example where the Hilbert space of wave functions is finite dimensional.
Basis representation
A wave function describes the state of a physical system Template:Ket, by expanding it in terms of other possible states of the same system – collectively referred to as a basis or representation Template:Ket. In what follows, all wave functions are assumed to be normalized.
An element of a vector space can be expressed in different bases elements; and so the same applies to wave functions. The components of a wave function describing the same physical state take different complex values depending on the basis being used; however, just like elements of a vector space, the wave function itself is not dependent on the basis chosen. Choosing a new coordinate system does not change the vector itself, only the representation of the vector with respect to the new coordinate frame, since the components will be different but the linear combination of them still equals the vector.
Finite dimensional Hilbert spaces
A wave function ψ with n components describes how to express the state of the physical system Template:Ket as the linear combination of n basis elements Template:Ket, (i = 1, 2...n). Following is a breakdown of the used formalism.
In bra–ket notation, the quantum state of a particle can be written as a ket;
- $|\; \backslash psi\; \backslash rangle$
= \sum_{i = 1}^n c_i | \varepsilon_i \rangle
= c_1 | \varepsilon_1 \rangle + c_2 | \varepsilon_2 \rangle + \cdots c_n | \varepsilon_n \rangle
= \begin{bmatrix} \langle \varepsilon_1 | \psi \rangle \\ \vdots \\ \langle \varepsilon_n | \psi \rangle \end{bmatrix}
= \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix} .
The basis here is orthonormal:
- $\backslash langle\; \backslash varepsilon\_i\; |\; \backslash varepsilon\_j\; \backslash rangle\; =\; \backslash delta\_\{ij\},$
where δ_{ij} is the Kronecker delta. The corresponding bra is the Hermitian conjugate – the transposed complex conjugate matrix (into a row matrix/row vector):
- $\backslash begin\{align\}$
\langle \psi | = | \psi \rangle^{*} & = \begin{bmatrix} \langle \varepsilon_1 | \psi \rangle & \cdots & \langle \varepsilon_n | \psi \rangle \end{bmatrix}^{*} = \begin{bmatrix} \langle \varepsilon_1 | \psi \rangle^{*} & \cdots & \langle \varepsilon_n | \psi \rangle^{*} \end{bmatrix} \\
& = \begin{bmatrix} c_1 & \cdots & c_n \end{bmatrix}^{*} = \begin{bmatrix} c_1^{*} & \cdots & c_n^{*} \end{bmatrix}
\end{align}
Kets are analogous to the more elementary Euclidean vectors, although the components are complex-valued. The state can be expanded in any convenient basis of the Hilbert space. Simple examples can be found from a two-state quantum system, two energy eigenstates:
- $|\; \backslash psi\; \backslash rangle\; =\; \backslash psi\_1\; |\; E\_1\; \backslash rangle\; +\; \backslash psi\_2\; |\; E\_2\; \backslash rangle.$
and two spin states (up or down):
- $|\; \backslash psi\; \backslash rangle\; =\; \backslash psi\_+\; |\; \backslash uparrow\_z\; \backslash rangle\; +\; \backslash psi\_\{-\}\; |\; \backslash downarrow\_z\; \backslash rangle\; ,$
(see below for details of this frequent case). In these examples, the particle is not in any one definite or preferred state, but rather in both at the same time – hence the term superposition. The relative chance of which state occurs is related to the (moduli squares of the) coefficients.
Projecting the initial state Template:Ket onto the particular state the system collapses to Template:Ket, gives the complex number;
- $\backslash begin\{align\}\; \backslash langle\; \backslash varepsilon\_q\; |\; \backslash psi\; \backslash rangle\; \&\; =\; \backslash langle\; \backslash varepsilon\_q\; |\; \backslash left\; (\; \backslash sum\_\{i\; =\; 1\}^n\; c\_i\; |\; \backslash varepsilon\_i\; \backslash rangle\; \backslash right\; )\; \backslash \backslash $
& = c_1 \langle \varepsilon_q | \varepsilon_1 \rangle
+ c_2 \langle \varepsilon_q | \varepsilon_2 \rangle
+ \cdots + c_q \langle \varepsilon_q | \varepsilon_q \rangle
+ \cdots c_n \langle \varepsilon_q | \varepsilon_n \rangle \\
& = c_q \,,
\end{align}
so the modulus squared of this gives a real number;
- $|c\_q|^2\; =\; \{\; |\; \backslash langle\; \backslash varepsilon\_q\; |\; \backslash psi\; \backslash rangle\; |\; \}^2\; \backslash ,,$
the probability of state Template:Ket occurring. The sum of the probabilities of all possible states must sum to 1 (see normalization using kets below), implying the constraint:
- $\backslash sum\_i\; |\; c\_i\; |^2\; =\; 1$
Closure relation in the discrete bases
Taking the state above
- $|\backslash psi\backslash rangle\; =\; \backslash sum\_\{i=1\}^n\; c\_i\; |\; \backslash varepsilon\_i\; \backslash rangle\; =\; \backslash sum\_\{i=1\}^n\; \backslash langle\; \backslash varepsilon\_i\; |\; \backslash psi\; \backslash rangle|\; \backslash varepsilon\_i\; \backslash rangle\; =\; \backslash left(\backslash sum\_\{i=1\}^n\; |\; \backslash varepsilon\_i\; \backslash rangle\; \backslash langle\; \backslash varepsilon\_i\; |\; \backslash right\; )\; |\; \backslash psi\; \backslash rangle\; \backslash ,$
we obtain the closure relation:
- $\backslash sum\_\{i=1\}^n\; |\; \backslash varepsilon\_i\; \backslash rangle\; \backslash langle\; \backslash varepsilon\_i\; |\; =\; 1\; .$
The equality to unity means this is an identity operator (its action on any state leaves it unchanged). Suppose we have another wavefunction in the same basis:
- $|\; \backslash chi\; \backslash rangle\; =\; \backslash sum\_\{j\; =\; 1\}^n\; z\_j\; |\; \backslash varepsilon\_j\; \backslash rangle\; =\; z\_1\; |\; \backslash varepsilon\_1\; \backslash rangle\; +\; z\_2\; |\; \backslash varepsilon\_2\; \backslash rangle\; +\; \backslash cdots\; z\_n\; |\; \backslash varepsilon\_n\; \backslash rangle\; =\; \backslash begin\{bmatrix\}\; \backslash langle\; \backslash varepsilon\_1\; |\; \backslash chi\; \backslash rangle\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; \backslash langle\; \backslash varepsilon\_n\; |\; \backslash chi\; \backslash rangle\; \backslash end\{bmatrix\}\; =\; \backslash begin\{bmatrix\}\; z\_1\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; z\_n\; \backslash end\{bmatrix\}\; .$
then the inner product can be obtained:
- $\backslash langle\; \backslash chi\; |\; \backslash psi\; \backslash rangle\; =\; \backslash langle\; \backslash chi\; |\; 1\; |\; \backslash psi\; \backslash rangle\; =\; \backslash langle\; \backslash chi\; |\; \backslash left(\; \backslash sum\_\{i=1\}^n\; |\; \backslash varepsilon\_i\; \backslash rangle\; \backslash langle\; \backslash varepsilon\_i\; |\; \backslash right)\; |\; \backslash psi\; \backslash rangle\; =\; \backslash sum\_\{i=1\}^n\; \backslash langle\; \backslash chi\; |\; \backslash varepsilon\_i\; \backslash rangle\; \backslash langle\; \backslash varepsilon\_i\; |\; \backslash psi\; \backslash rangle\; =\; \backslash sum\_\{i=1\}^n\; z\_i^\{*\}\; c\_i.$
Normalization in discrete bases
The norm or magnitude of the state vector Template:Ket is:
- $\backslash |\backslash psi\backslash |^2\; =\; \backslash langle\; \backslash psi\; |\; \backslash psi\; \backslash rangle\; =\; \backslash sum\_\{j=1\}^n\; |\; c\_j\; |^2\; .$
which says the projection of a complex probability amplitude onto itself is real. The sum of all probabilities of basis states occurring must be unity:
- $\backslash frac\{1\}\{\backslash |\backslash psi\backslash |^2\}\backslash langle\; \backslash psi\; |\; \backslash psi\; \backslash rangle\; =\; \backslash frac\{1\}\{\backslash |\backslash psi\backslash |^2\}\backslash sum\_\{j=1\}^n\; |\; c\_j\; |^2\; =\; 1\; \backslash ,,$
so the normalized state Template:Ket in all generality is:
- $|\; \backslash psi\_N\; \backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{\backslash langle\; \backslash psi|\backslash psi\backslash rangle\}\}\; |\; \backslash psi\; \backslash rangle$
Compare the similarity with Euclidean unit vectors a in elementary vector calculus:
- $\backslash mathbf\{\backslash hat\{a\}\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{\backslash mathbf\{a\}\backslash cdot\backslash mathbf\{a\}\}\}\backslash mathbf\{a\}$
The parallels are identical: the magnitude of the vector, geometric or abstract, is reduced to 1 by dividing by its magnitude.
Application to one spin-½ particle (neglect spatial freedom)
A simple and important case is a spin-½ particle, but for this instance ignore its spatial degrees of freedom. Using the definition above, the wave function can now be written without position dependence:
- $\backslash Psi\; \backslash left\; (\; s\_z,t\; \backslash right\; )$,
where again s_{z} is the spin quantum number in the z-direction, either +1/2 or −1/2. So at a given time t, Ψ is completely characterized by just the two complex numbers Ψ(+1/2,t) and Ψ(–1/2,t). For simplicity these are often written as Ψ(+1/2,t) ≡ Ψ_{+} ≡ Ψ_{↑}, and Ψ(–1/2,t) ≡ Ψ_{–} ≡ Ψ_{↓} respectively. This is still called a "wave function", even though in this situation it has no resemblance to familiar waves (like mechanical waves), being only a pair of numbers instead of a continuous function.
Using the above formalism, the two numbers characterizing the wave function can be written as a column vector:
- $\backslash vec\; \backslash psi\; =\; \backslash begin\{bmatrix\}\; c\_1\; \backslash \backslash \; c\_2\; \backslash end\{bmatrix\}$
where c_{1} = Ψ_{+} and c_{2} = Ψ_{−}. Therefore the set of all possible wave functions is a two dimensional complex vector space. If the particle's spin projection in the z-direction is measured, it will be spin up (+1/2 ≡ ↑_{z}) with probability |c_{1}|^{2}, and spin down (–1/2 ≡ ↓_{z}) with probability |c_{2}|^{2}.
In bra–ket notation this can be written:
- $\backslash begin\{align\}\; |\; \backslash psi\; \backslash rangle\; \&\; =\; c\_1\; |\; \backslash uparrow\_z\; \backslash rangle\; +\; c\_2\; |\; \backslash downarrow\_z\; \backslash rangle\; \backslash \backslash $
& = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix} = \begin{bmatrix} \Psi_{+} \\ \Psi_{-} \end{bmatrix} = \begin{bmatrix} \langle \uparrow_z | \psi \rangle \\ \langle \downarrow_z | \psi \rangle \end{bmatrix}
\end{align},
using the basis vectors (in alternate notations)
- $|\; \backslash uparrow\_z\; \backslash rangle\; \backslash equiv\; |\; +\; \backslash rangle$ for "spin up" or s_{z} = +1/2,
$|\; \backslash downarrow\_z\; \backslash rangle\; \backslash equiv\; |\; -\; \backslash rangle$ for "spin down" or s_{z} = –1/2.
The normalization requirement is
- $|c\_1|^2+|c\_2|^2\; =\; 1,$
which says the probability of the particle in the spin up state (↑_{z}, corresponding to the coefficient c_{1}) plus the probability in the spin down (↓_{z}, corresponding to the coefficient c_{2}) state is 1.
To see this explicitly for this case, expand the ket in terms of the bases:
- $|\; \backslash psi\; \backslash rangle\; =\; c\_1|\; \backslash uparrow\_z\; \backslash rangle\; +\; c\_2|\; \backslash downarrow\_z\; \backslash rangle\; ,$
implying
- $\backslash langle\; \backslash psi\; |\; =\; c\_1^\{*\}\; \backslash langle\; \backslash uparrow\_z\; |\; +\; c\_2^\{*\}\; \backslash langle\; \backslash downarrow\_z\; |\; ,$
taking the inner product (and recalling orthonormality) leads to the normalization condition:
- $\backslash begin\{align\}\; \backslash langle\; \backslash psi\; |\; \backslash psi\; \backslash rangle\; \&\; =\; \backslash left\; (\; c\_1|\; \backslash uparrow\_z\; \backslash rangle\; +\; c\_2|\; \backslash downarrow\_z\; \backslash rangle\; \backslash right\; )\; \backslash left\; (\; c\_1^\{*\}\; \backslash langle\; \backslash uparrow\_z\; |\; +\; c\_2^\{*\}\; \backslash langle\; \backslash downarrow\_z\; |\; \backslash right\; )\; \backslash \backslash $
& = c_1| \uparrow_z \rangle \left ( c_1^{*} \langle \uparrow_z | + c_2^{*} \langle \downarrow_z | \right ) + c_2| \downarrow_z \rangle \left ( c_1^{*} \langle \uparrow_z | + c_2^{*} \langle \downarrow_z | \right ) \\
& = c_1 c_1^{*} \langle \uparrow_z | \uparrow_z \rangle + c_1 c_2^{*} \langle \downarrow_z | \uparrow_z \rangle + c_2 c_1^{*} \langle \uparrow_z | \downarrow_z \rangle + c_2 c_2^{*} \langle \downarrow_z | \downarrow_z \rangle \\
& = |c_1|^2+|c_2|^2 \\
& = 1
\end{align}
Infinite dimensional vectors
States can have countably infinitely many components;
- $\backslash left\; |\; \backslash psi\; \backslash right\; \backslash rangle$
= \sum_{i = 1}^\infty c_i \left | \varepsilon_i \right \rangle
= c_1 \left | \varepsilon_1 \right \rangle + c_2 \left | \varepsilon_2 \right \rangle + \cdots
= \begin{bmatrix} \left \langle \varepsilon_1 | \psi \right \rangle \\ \vdots \\ \left \langle \varepsilon_n | \psi \right \rangle \\ \vdots \end{bmatrix}
= \begin{bmatrix} c_1 \\ \vdots \\ c_n \\ \vdots \end{bmatrix} .
with corresponding bra as before:
- $\backslash begin\{align\}$
\langle \psi | = | \psi \rangle^{*} & = \begin{bmatrix} \langle \varepsilon_1 | \psi \rangle & \cdots & \langle \varepsilon_n | \psi \rangle & \cdots \end{bmatrix}^{*} = \begin{bmatrix} \langle \varepsilon_1 | \psi \rangle^{*} & \cdots & \langle \varepsilon_n | \psi \rangle^{*} & \cdots \end{bmatrix} \\
& = \begin{bmatrix} c_1 & \cdots & c_n & \cdots \end{bmatrix}^{*} = \begin{bmatrix} c_1^{*} & \cdots & c_n^{*} & \cdots \end{bmatrix}
\end{align}
They can also have an uncountably infinite number of components. The collection of all states Template:Ket is a continuum of states. While finite or countably infinite basis vectors are summed over a discrete index, uncountably infinite basis vectors are integrated over a continuous index (a variable of a function). In what follows, all integrals are with respect to the basis variable ε (a real number or vector, not complex-valued), over the required range. Usually this is just the real line or subset intervals of it. The state Template:Ket is given by:
- $|\; \backslash psi\; \backslash rangle\; =\; \backslash int\; d\; \backslash varepsilon\; |\; \backslash varepsilon\; \backslash rangle\; \backslash psi(\backslash varepsilon)\; \backslash ,,$
with corresponding bra:
- $\backslash langle\; \backslash psi\; |\; =\; \backslash int\; d\; \backslash varepsilon\; \backslash langle\; \backslash varepsilon\; |\; \{\backslash psi(\backslash varepsilon)\}^\{*\}\; \backslash ,,$
and again the basis here is orthonormal:
- $\backslash langle\; \backslash varepsilon\; |\; \backslash varepsilon\text{'}\; \backslash rangle\; =\; \backslash delta\; (\backslash varepsilon-\backslash varepsilon\text{'})\backslash ,.$
As with the discrete bases, a symbol ε is used in the basis states, two common notations are Template:Ket and sometimes Template:Ket. A particular basis ket may be subscripted Template:Ket ≡ Template:Ket or primed Template:Ket ≡ Template:Ket.
The components of the state Template:Ket are still Template:Bra-ket, the projection of the state onto some basis is a function;
- $\backslash langle\; \backslash varepsilon\_0\; |\; \backslash psi\; \backslash rangle\; =\; \backslash langle\; \backslash varepsilon\_0\; |\; \backslash left(\; \backslash int\; d\; \backslash varepsilon\; |\; \backslash varepsilon\; \backslash rangle\; \backslash psi(\backslash varepsilon)\; \backslash right)\; =\; \backslash int\; d\; \backslash varepsilon\; \backslash langle\; \backslash varepsilon\_0\; |\; \backslash varepsilon\; \backslash rangle\; \backslash psi(\backslash varepsilon)\; =\; \backslash int\; d\; \backslash varepsilon\; \backslash delta(\; \backslash varepsilon\_0\; -\; \backslash varepsilon\; )\; \backslash psi(\backslash varepsilon)\; =\; \backslash psi(\backslash varepsilon\_0)\; \backslash ,,$
This time
- $|\; \backslash psi(\backslash varepsilon)\; |^2\; =\; |\; \backslash langle\; \backslash varepsilon\; |\; \backslash psi\; \backslash rangle\; |^2$
is the probability density function of measuring the observable ε, so integrating this with respect to ε between a ≤ ε ≤ b gives:
- $P\_\{a\; \backslash leq\; \backslash varepsilon\; \backslash leq\; b\}\; =\; \backslash int\_a^b\; d\backslash varepsilon\; |\; \backslash psi(\backslash varepsilon)\; |^2\; =\; \backslash int\_a^b\; d\backslash varepsilon|\; \backslash langle\; \backslash varepsilon\; |\; \backslash psi\; \backslash rangle\; |^2\; \backslash ,,$
the probability of finding the system with ε between ε = a and ε = b.
Closure relation in continuous bases
Taking the state above
- $|\backslash psi\backslash rangle\; =\; \backslash int\; d\; \backslash varepsilon\; \backslash ,\; |\; \backslash varepsilon\; \backslash rangle\; \backslash psi(\backslash varepsilon)\; =\; \backslash int\; d\; \backslash varepsilon\; \backslash ,\; |\; \backslash varepsilon\; \backslash rangle\; \backslash langle\; \backslash varepsilon\; |\; \backslash psi\; \backslash rangle\; =\; \backslash left(\backslash int\; d\backslash varepsilon\; \backslash ,\; |\; \backslash varepsilon\; \backslash rangle\; \backslash langle\; \backslash varepsilon\; |\; \backslash right)\; |\; \backslash psi\; \backslash rangle\; \backslash ,$
we obtain the closure relation:
- $\backslash int\; d\backslash varepsilon\; \backslash ,\; |\; \backslash varepsilon\; \backslash rangle\; \backslash langle\; \backslash varepsilon\; |\; =\; 1\; \backslash ,$
Also the inner product can be obtained:
- $\backslash langle\; \backslash chi\; |\; \backslash psi\; \backslash rangle\; =\; \backslash langle\; \backslash chi\; |\; 1\; |\; \backslash psi\; \backslash rangle\; =\; \backslash langle\; \backslash chi|\; \backslash left(\; \backslash int\; d\; \backslash varepsilon\; |\; \backslash varepsilon\; \backslash rangle\; \backslash langle\; \backslash varepsilon\; |\; \backslash right)\; |\; \backslash psi\; \backslash rangle\; =\; \backslash int\; d\; \backslash varepsilon\; \backslash langle\; \backslash chi\; |\; \backslash varepsilon\; \backslash rangle\; \backslash langle\; \backslash varepsilon\; |\; \backslash psi\; \backslash rangle\; =\; \backslash int\; d\; \backslash varepsilon\; \backslash chi(\backslash varepsilon)^\{*\}\; \backslash psi(\backslash varepsilon)\; .$
Normalization in continuous bases
Taking the inner product;
- $\backslash langle\; \backslash psi\; |\; \backslash psi\; \backslash rangle\; =\; \backslash int\; d\; \backslash varepsilon\; \backslash ,\; |\; \backslash psi(\backslash varepsilon)\; |^2\; =\; \backslash |\backslash psi\backslash |^2\; .$
This integral is the total probability of all basis states occurring, so it must be 1 as before:
- $\backslash frac\{1\}\{\backslash |\backslash psi\backslash |^2\}\backslash langle\; \backslash psi\; |\; \backslash psi\; \backslash rangle\; =\; \backslash frac\{1\}\{\backslash |\backslash psi\backslash |^2\}\; \backslash int\; d\; \backslash varepsilon\; |\; \backslash psi(\backslash varepsilon)\; |^2\; =\; 1$
hence
- $|\; \backslash psi\_N\; \backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{\backslash langle\; \backslash psi|\backslash psi\backslash rangle\}\}\; |\; \backslash psi\; \backslash rangle$
Kets are much easier to normalize than the above procedure; solving the equation after evaluating the normalizing integral.
Application to position, momentum and spin state spaces
The following are illustrated in position space. For the momentum space, the equations need only the replacement x → p_{x} in 1d or r → p in 3d. Of course, they can be generalized for more than one particle, requiring multiple sums or integrals for each particle, as shown previously.
One spin-0 particle in one dimension
For a spinless particle in one spatial dimension (the x-axis or real line), the state Template:Ket can be expanded in terms of a continuum of states; i.e. Template:Ket ≡ Template:Ket → Template:Ket ≡ Template:Ket, corresponding to each x.
If the particle is confined to a region R (a subset of the x-axis), the state is:
- $|\; \backslash psi\; \backslash rangle\; =\; \backslash int\backslash limits\_R\; d\; x\; \backslash ,\; |\; x\; \backslash rangle\; \backslash langle\; x\; |\; \backslash psi\; \backslash rangle\; =\; \backslash int\backslash limits\_R\; d\; x\; \backslash ,\; \backslash psi(x)\; |\; x\; \backslash rangle$
leading to the closure relation
- $1\; =\; \backslash int\backslash limits\_R\; d\; x\; \backslash ,\; |\; x\; \backslash rangle\; \backslash langle\; x\; |$
and the inner product as stated at the beginning of this article (in that case R = (−∞, ∞)):
- $\backslash langle\; \backslash chi\; |\; \backslash psi\; \backslash rangle\; =\; \backslash int\backslash limits\_R\; d\; x\; \backslash ,\; \backslash langle\; \backslash chi\; |\; x\; \backslash rangle\; \backslash langle\; x\; |\; \backslash psi\; \backslash rangle\; =\; \backslash int\backslash limits\_R\; d\; x\; \backslash ,\; \backslash chi(x)^\{*\}\; \backslash psi(x)\; \backslash ,.$.
The "wavefunction" described previously is simply a component of the complex state vector. Projecting Template:Ket onto a particular position state Template:Ket, where x_{0} is in R:
- $\backslash langle\; x\_0\; |\; \backslash psi\; \backslash rangle\; =\; \backslash int\backslash limits\_R\; d\; x\; \backslash ,\; \backslash langle\; x\_0\; |\; x\; \backslash rangle\; \backslash psi(x)\; =\; \backslash int\backslash limits\_R\; d\; x\; \backslash ,\; \backslash delta(\; x\_0\; -\; x\; )\; \backslash psi(x)\; =\; \backslash psi(x\_0)\; \backslash ,.$
One spin-0 particle in three dimensions
The generalization of the previous result is straightforward. In three dimensions, Template:Ket can be expanded in terms of a continuum of states with definite position, so Template:Ket ≡ Template:Ket → Template:Ket ≡ Template:Ket ≡ Template:Ket, corresponding to each r = (x, y, z).
If the particle is confined to a region R (a subset of 3d space), the state is;
- $|\; \backslash psi\; \backslash rangle\; =\; \backslash int\backslash limits\_R\; d^3\backslash mathbf\{r\}\; \backslash ,\; |\; \backslash mathbf\{r\}\; \backslash rangle\; \backslash langle\; \backslash mathbf\{r\}\; |\; \backslash psi\backslash rangle\; =\; \backslash int\backslash limits\_R\; d^3\backslash mathbf\{r\}\; \backslash ,\; \backslash psi(\backslash mathbf\{r\})\; |\; \backslash mathbf\{r\}\; \backslash rangle$
The closure relation is
- $1\; =\; \backslash int\backslash limits\_R\; d^3\backslash mathbf\{r\}\; \backslash ,\; |\; \backslash mathbf\{r\}\; \backslash rangle\; \backslash langle\; \backslash mathbf\{r\}\; |$
leading to the inner product of Template:Ket with itself leads to the normalization conditions in the three-dimensional definitions above:
- $\backslash langle\; \backslash chi\; |\; \backslash psi\; \backslash rangle\; =\; \backslash int\backslash limits\_R\; d^3\backslash mathbf\{r\}\; \backslash ,\; \backslash langle\; \backslash chi\; |\; \backslash mathbf\{r\}\; \backslash rangle\; \backslash langle\; \backslash mathbf\{r\}\; |\; \backslash psi\; \backslash rangle\; =\; \backslash int\backslash limits\_R\; d^3\backslash mathbf\{r\}\; \backslash ,\; \backslash chi(\backslash mathbf\{r\})^\{*\}\; \backslash psi(\backslash mathbf\{r\})$.
Projecting $|\; \backslash psi\; \backslash rangle$ onto a particular position state Template:Ket, where r_{0} is in R:
- $\backslash langle\; \backslash mathbf\{r\}\_0\; |\; \backslash psi\; \backslash rangle\; =\; \backslash int\backslash limits\_R\; d^3\; \backslash mathbf\{r\}\; \backslash ,\; \backslash langle\; \backslash mathbf\{r\}\_0\; |\; \backslash mathbf\{r\}\; \backslash rangle\; \backslash psi(\backslash mathbf\{r\})\; =\; \backslash int\backslash limits\_R\; d^3\; \backslash mathbf\{r\}\; \backslash ,\; \backslash delta(\; \backslash mathbf\{r\}\_0\; -\; \backslash mathbf\{r\}\; )\; \backslash psi(\backslash mathbf\{r\})\; =\; \backslash psi(\backslash mathbf\{r\}\_0)$
The above expressions take the same form for any number of spatial dimensions.
One spin particle in three dimensions
For a particle with spin s, in all three spatial dimensions, the basis states Template:Ket are a combination of the discrete variable s_{z} (the z-component spin quantum number) and the continuous variable r (position of the particle).^{[12]} Applying the above formalism, the state can be written:
- $|\; \backslash Psi\; \backslash rangle\; =\; \backslash sum\_\{s\_z\}\; \backslash int\backslash limits\_R\; d^3\; \backslash ,\; \backslash mathbf\{r\}\; \backslash Psi(\backslash mathbf\{r\},s\_z)\; |\; \backslash mathbf\{r\},\; s\_z\; \backslash rangle$
and therefore the closure relation (identity operator) is:
- $1\; =\; \backslash sum\_\{s\_z\}\; \backslash int\backslash limits\_R\; d^3\; \backslash ,\; \backslash mathbf\{r\}\; |\; \backslash mathbf\{r\},s\_z\backslash rangle\; \backslash langle\; \backslash mathbf\{r\}\; ,\; s\_z\; |$
Projecting Ψ onto a particular position-spin state Template:Ket, where r_{0} is in R:
- $\backslash langle\; \backslash mathbf\{r\}\_0,\; m\; |\; \backslash Psi\; \backslash rangle\; =\; \backslash sum\_\{s\_z\}\backslash int\backslash limits\_R\; d^3\; \backslash mathbf\{r\}\; \backslash ,\; \backslash langle\; \backslash mathbf\{r\}\_0,\; m\; |\; \backslash mathbf\{r\},\; s\_z\; \backslash rangle\; \backslash Psi(\backslash mathbf\{r\},\; s\_z)\; =\; \backslash sum\_\{s\_z\}\backslash int\backslash limits\_R\; d^3\; \backslash mathbf\{r\}\; \backslash ,\; \backslash delta\_\{m\; \backslash ,\; s\_z\}\backslash delta(\; \backslash mathbf\{r\}\_0\; -\; \backslash mathbf\{r\}\; )\; \backslash Psi(\backslash mathbf\{r\},\; s\_z)\; =\; \backslash Psi(\backslash mathbf\{r\}\_0,\; m)\; \backslash ,.$
where the joint orthogonality relation
- $\backslash langle\; \backslash mathbf\{r\}\_0,\; m\; |\; \backslash mathbf\{r\},\; s\_z\; \backslash rangle\; =\; \backslash delta\_\{m\backslash ,s\_z\}\backslash delta(\; \backslash mathbf\{r\}\_0\; -\; \backslash mathbf\{r\}\; )$
has been used.
Time dependence
In the Schrödinger picture, the states evolve in time, so the time dependence is placed in Template:Ket according to:^{[13]}
- $|\backslash psi(t)\backslash rangle\; =\; \backslash sum\_i\; \backslash ,\; |\; \backslash varepsilon\_i\; \backslash rangle\; \backslash langle\; \backslash varepsilon\_i\; |\; \backslash psi(t)\backslash rangle\; =\; \backslash sum\_i\; c\_i(t)\; |\; \backslash varepsilon\; \backslash rangle$
for discrete bases, or
- $|\backslash psi(t)\backslash rangle\; =\; \backslash int\; d\backslash varepsilon\; \backslash ,\; |\; \backslash varepsilon\; \backslash rangle\; \backslash langle\; \backslash varepsilon\; |\; \backslash psi(t)\backslash rangle\; =\; \backslash int\; d\backslash varepsilon\; \backslash ,\; \backslash psi(\backslash varepsilon,t)\; |\; \backslash varepsilon\; \backslash rangle$
for continuous bases. However, in the Heisenberg picture the states Template:Ket are constant in time and time dependence is placed in the Heisenberg operators, so Template:Ket is not written as Template:Ket.
Wave function collapse
The physical meaning of the components of Template:Ket is given by the wave function collapse postulate also known as Wave function collapse. If the observable(s) ε (momentum and/or spin, position and/or spin, etc.) corresponding to states Template:Ket has distinct and definite values, λ_{i}, and a measurement of that variable is performed on a system in the state Template:Ket then the probability of measuring λ_{i} is |Template:Bra-ket|^{2}. If the measurement yields λ_{i}, the system "collapses" to the state Template:Ket, irreversibly and instantaneously.
Ontology
Whether the wave function really exists, and what it represents, are major questions in the interpretation of quantum mechanics. Many famous physicists of a previous generation puzzled over this problem, such as Schrödinger, Einstein and Bohr. Some advocate formulations or variants of the Copenhagen interpretation (e.g. Bohr, Wigner and von Neumann) while others, such as Wheeler or Jaynes, take the more classical approach^{[14]} and regard the wave function as representing information in the mind of the observer, i.e. a measure of our knowledge of reality. Some, including Schrödinger, Einstein, Bohm and Everett and others, argued that the wave function must have an objective, physical existence. The latter argument is consistent with the fact that whenever two observers both think that a system is in a pure quantum state, they will always agree on exactly what state it is in (but this may not be true if one or both of them thinks the system is in a mixed state).^{[15]} For more on this topic, see Interpretations of quantum mechanics.
Examples
Here are examples of wavefunctions for specific applications:
See also
References
2.Quantum Mechanics (Non-Relativistic Theory), L.D. Landau and E.M. Lifshitz, ISBN 0-08-020940-8
Further reading
External links
- [4]
- [5] Normalization.
- [6] Quantum Mechanics and Quantum Computation at BerkeleyXes:Función de onda normalizable
ko:파동함수
ja:規格化
fi:Normitettu aaltofunktio
zh:歸一條件
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