Wave function

Some trajectories of a harmonic oscillator (a ball attached to a spring) in classical mechanics (A–B) and quantum mechanics (C–H). In quantum mechanics (C–H), the ball has a wave function, which is shown with real part in blue and imaginary part in red. The trajectories C,D,E,F, (but not G or H) are examples of standing waves, (or "stationary states"). Each standing-wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have any energy.

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, |ψ|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 unit of measurement for ψ depends on the system. For one particle in three dimensions, its units are [length]–3/2. These unusual units are required so that an integral of |ψ|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 and can be found 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:

$\psi(\mathbf{r}) \leftrightarrow |\psi\rangle$

where |ψ 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 Ψ1(x) and Ψ2(x) can be defined as

$\langle \Psi_1 | \Psi_2 \rangle \equiv \int\limits_{-\infty}^\infty d x \, \Psi_1^*(x)\Psi_2(x) ,$

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 |ψ and |ϕ are two allowed states, then a|ψ + b|ϕ 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 { | εi } and orthonormal, that is

$\langle \varepsilon_i | \varepsilon_j \rangle = \delta_{ij},$

then an arbitrary vector |ψ can be expressed as

$| \psi \rangle = \sum_i c_i | \varepsilon_i \rangle,$

where the components are the (complex) numbers ci = εi|ψ 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,

$\langle \varepsilon | \varepsilon_0 \rangle = \delta \left ( \varepsilon - \varepsilon_0 \right ),$

then an arbitrary vector $| \psi \rangle$ can be expressed as

$| \psi \rangle = \int d \varepsilon \psi(\varepsilon) | \varepsilon \rangle .$

where the components are the functions ψ(ε) = ε|ψ This wave function is known as a continuous spectrum, since the bases are continuous.

Paramount to the analysis is the Kronecker delta, δij, and the Dirac delta function, δ(εε0), since the bases used are orthonormal. More detailed discussion of wave functions as elements of vector spaces is below, following further definitions.

Requirements

Continuity of the wavefunction and its first spatial derivative (in the x direction, y and z coordinates not shown), at some time t.

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.

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:

$\Psi(x,t)$,

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 axb) is:

$P_{a\le x\le b} = \int\limits_a^b d x\,|\Psi(x,t)|^2$

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:

$\int\limits_{-\infty}^\infty d x \, |\Psi(x,t)|^2 = 1$,

because if the particle is measured, there is 100% probability that it will be somewhere.

Momentum-space wavefunction

The particle also has a wave function in momentum space:

$\Phi(p,t)$

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:

$P_{a\le p\le b} = \int\limits_a^b d p \, |\Phi(p,t)|^2$,

analogous to the position case.

The normalization condition is also similar:

$\int\limits_{-\infty}^{\infty} d p \, \left | \Phi \left ( p, t \right ) \right |^2 = 1.$

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]

\begin{align} \Phi(p,t) & = \frac{1}{\sqrt{2\pi\hbar}}\int\limits_{-\infty}^\infty d x \, e^{-ipx/\hbar} \Psi(x,t)\\ &\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

$p = \hbar k,$

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:

\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

$\int\limits_{-\infty}^{\infty} dx \, |\Psi|^2 = 1 .$

From Ψ, we have |Ψ|2;

$| \Psi | ^2 = A^2 e^{i(kx - \omega t)} e^{-i(kx - \omega t)} =A^2 ,$

so the integral becomes;

$\int\limits_{-\infty}^0 dx \cdot 0 + \int\limits_0^L dx \, A^2 + \int\limits_L^\infty dx \cdot 0 = 1 ,$

therefore the constant is;

$A^2 L = 1 \rightarrow A = \frac{1}{\sqrt{L}} .$

The normalized wave function (in the region) is then given by;

$\Psi (x,t) = \frac{1}{\sqrt{L}} e^{i(kx-\omega t)}, \quad 0 \leq x \leq L.$

Definition (other cases)

Many spin-0 particles in one spatial dimension

Travelling waves of two free particles. Top is position space wavefunction, bottom is momentum space wavefunction, with corresponding probability densities.

The previous wavefunction can be generalized to incorporate N particles in one dimension:

$\Psi(x_1,x_2,\cdots x_N, t)$,

The probability that particle 1 is in an x-interval R1 = [a1,b1] and particle 2 in interval R2 = [a2,b2] etc., up to particle N in interval RN = [aN,bN], all measured simultaneously at time t, is given by:

$P_{x_1\in R_1,x_2\in R_2 \cdots x_N\in R_N} = \int\limits_{a_1}^{b_1} d x_1 \int\limits_{a_2}^{b_2} d x_2 \cdots \int\limits_{a_N}^{b_N} d x_N | \Psi(x_1 \cdots x_N,t)|^2$

The normalization condition becomes:

$\int\limits_{-\infty}^\infty d x_1 \int\limits_{-\infty}^\infty d x_2 \cdots \int\limits_{-\infty}^\infty d x_N |\Psi(x_1 \cdots x_N,t)|^2 = 1$.

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 electron probability density for the first few hydrogen atom electron orbitals shown as cross-sections. These orbitals form an orthonormal basis for the wave function of the electron. Different orbitals are depicted with different scale.

The position-space wave function of a single particle in three spatial dimensions is similar to the case of one spatial dimension above:

$\Psi(\mathbf{r},t)$

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:

$P_{\mathbf{r}\in R} = \int\limits_R d^3\mathbf{r} \, \left |\Psi(\mathbf{r},t) \right |^2$

(a three-dimensional integral over the region R, with differential volume element d3r, also written "dV" or "dx dy dz"). The normalization condition is:

$\int\limits_{{\rm all \, space}} \left | \Psi(\mathbf{r},t)\right |^2 d ^3\mathbf{r} = 1,$

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:

$\Phi(\mathbf{p},t)$

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 px between a and b, py between c and d, and pz between e and f, is given by:

$P_{p_x\in[a,b],p_y\in[c,d],p_z\in[e,f]} = \int\limits_e^f dp_z\,\int\limits_c^d d p_y \, \int\limits_a^b d p_x \, \left | \Phi \left ( \mathbf{p}, t \right ) \right |^2 ,$

hence the normalization:

$\int\limits_{{\rm all \, space}} d^3\mathbf{p} \, \left | \Phi \left ( \mathbf{p}, t \right ) \right |^2 = 1.$

analogous to space, d3p = dpxdpydpz is a differential 3-momentum volume element in momentum space.

Relation between wavefunctions

The generalization of the previous Fourier transform is[10]

\begin{align} \Phi(\mathbf{p},t) & = \frac{1}{\sqrt{\left(2\pi\hbar\right)^3}}\int\limits_{{\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_{{\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]

$\Psi(\mathbf{r}_1,\mathbf{r}_2 \cdots \mathbf{r}_N,t)$

where ri 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 R1 and particle 2 is in region R2 and so on is:

$P_{\mathbf{r}_1\in R_1,\mathbf{r}_2\in R_2 \cdots \mathbf{r}_N\in R_N} = \int\limits_{R_1} d ^3\mathbf{r}_1 \int\limits_{R_2} d ^3\mathbf{r}_2\cdots \int\limits_{R_N} d ^3\mathbf{r}_N |\Psi(\mathbf{r}_1 \cdots \mathbf{r}_N,t)|^2$

The normalization condition is:

$\int\limits_{{\rm all \, space}} d ^3\mathbf{r}_1 \int\limits_{{\rm all \, space}} d ^3\mathbf{r}_2\cdots \int\limits_{{\rm all \, space}} d ^3\mathbf{r}_N |\Psi(\mathbf{r}_1 \cdots \mathbf{r}_N,t)|^2 = 1$

(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]

$\Psi = \phi(t)\prod_{i=1}^N\psi(\mathbf{r}_i) = \phi(t)\psi(\mathbf{r}_1)\psi(\mathbf{r}_2)\cdots\psi(\mathbf{r}_N).$

One particle with spin in three dimensions

For a particle with spin, the wave function can be written in "position-spin-space" as:

$\Psi(\mathbf{r},s_z,t)$

where r is a position in three-dimensional space, t is time, and sz 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 sz parameter, unlike r and t, is a discrete variable. For example, for a spin-1/2 particle, sz can only be +1/2 or −1/2, and not any other value. (In general, for spin s, sz 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 R1 and its spin projection quantum number is a certain value sz = m is:

$P_{\mathbf{r}\in R,s_z=m} = \int\limits_{R} d ^3\mathbf{r} |\Psi(\mathbf{r},t,m)|^2$

The normalization condition is:

$\sum_{\mathrm{all\, }s_z} \int\limits_{{\rm all \, space}} |\Psi(\mathbf{r},t,s_z)|^2 d ^3\mathbf{r} = 1$.

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:

$\Psi(\mathbf{r}_1, \mathbf{r}_2 \cdots \mathbf{r}_N, s_{z\,1}, s_{z\,2} \cdots s_{z\,N}, t)$

The probability that particle 1 is in region R1 with spin sz1 = m1 and particle 2 is in region R2 with spin sz2 = m2 etc. reads (probability subscripts now removed due to their great length):

$P = \int\limits_{R_1} d ^3\mathbf{r}_1 \int\limits_{R_2} d ^3\mathbf{r}_2\cdots \int\limits_{R_N} d ^3\mathbf{r}_N \left | \Psi\left (\mathbf{r}_1 \cdots \mathbf{r}_N,m_1\cdots m_N,t \right ) \right |^2$

The normalization condition is:

$\sum_{s_{z\,N}} \cdots \sum_{s_{z\,2}} \sum_{s_{z\,1}} \int\limits_{{\rm all \, space}} d ^3\mathbf{r}_1 \int\limits_{{\rm all \, space}} d ^3\mathbf{r}_2\cdots \int\limits_{{\rm all \, space}} d ^3 \mathbf{r}_N \left | \Psi \left (\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) \right |^2 = 1$

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]

$\Psi = \phi(t)\prod_{i=1}^N\psi(\mathbf{r}_i,s_{z\,i}) = \phi(t)\psi(\mathbf{r}_1,s_{z\,1})\psi(\mathbf{r}_2,s_{z\,2})\cdots\psi(\mathbf{r}_N,s_{z\,N}).$

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:

$\Psi \left ( \mathbf{r},\mathbf{r'},\mathbf{r}_3,\mathbf{r}_4,\cdots \right ) = \left ( -1 \right )^{2s} \Psi \left ( \mathbf{r'},\mathbf{r},\mathbf{r}_3,\mathbf{r}_4,\cdots \right )$

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.

Units of the wavefunction

Even though wavefunctions are complex numbers, both the real and imaginary parts each have the same units (the imaginary unit i is a number without unit). The units of ψ depend on the number of particles the wavefunction describes, and the number of spatial or momentum dimensions of the system. In general, for N particles with positions r1, r2, ..., rN in n spatial dimensions, the normalization conditions require ψ to have units of [length]Nn/2. The square root of the length unit is removed when one finds |ψ|2, which has units of [length]Nn.

In momentum space, length is replaced by momentum, and the units are [momentum]Nn/2.

These results are true for particles with or without spin, since for particles with spin, the summations are over dimensionless spin quantum numbers.

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:

$\hat{H} (A\Psi) = i\hbar\frac{\partial }{\partial t}(A\Psi) \rightarrow \hat{H} \Psi = i\hbar\frac{\partial }{\partial t}\Psi$

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 Ak of a complex vector |A = ∑k Ak|ek, which belongs to a countably infinite-dimensional Hilbert space; there are countably infinitely many k values and basis vectors |ek.
Continuous components ψ(x) of a complex vector |ψ = ∫dx ψ(x)|ψ, which belongs to an uncountably infinite-dimensional Hilbert space; there are uncountably infinitely many x values and basis vectors |x.
Components of complex vectors plotted against index number; discrete k and continuous x. Two probability amplitudes out of infinitely many are highlighted.

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 |ψ, by expanding it in terms of other possible states of the same system – collectively referred to as a basis or representation |εi. 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 |ψ as the linear combination of n basis elements |εi, (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;

$| \psi \rangle = \sum_{i = 1}^n c_i | \varepsilon_i \rangle = c_1 | \varepsilon_1 \rangle + c_2 | \varepsilon_2 \rangle + \cdots c_n | \varepsilon_n \rangle = \begin{bmatrix} \langle \varepsilon_1 | \psi \rangle \\ \vdots \\ \langle \varepsilon_n | \psi \rangle \end{bmatrix} = \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix} .$

The basis here is orthonormal:

$\langle \varepsilon_i | \varepsilon_j \rangle = \delta_{ij},$

where δij is the Kronecker delta. The corresponding bra is the Hermitian conjugate – the transposed complex conjugate matrix (into a row matrix/row vector):

\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:

$| \psi \rangle = \psi_1 | E_1 \rangle + \psi_2 | E_2 \rangle.$

and two spin states (up or down):

$| \psi \rangle = \psi_+ | \uparrow_z \rangle + \psi_{-} | \downarrow_z \rangle ,$

(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 |ψ onto the particular state the system collapses to |ε, gives the complex number;

\begin{align} \langle \varepsilon_q | \psi \rangle & = \langle \varepsilon_q | \left ( \sum_{i = 1}^n c_i | \varepsilon_i \rangle \right ) \\ & = 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;

$|c_q|^2 = { | \langle \varepsilon_q | \psi \rangle | }^2 \,,$

the probability of state |εq occurring. The sum of the probabilities of all possible states must sum to 1 (see normalization using kets below), implying the constraint:

$\sum_i | c_i |^2 = 1$

Closure relation in the discrete bases

Taking the state above

$|\psi\rangle = \sum_{i=1}^n c_i | \varepsilon_i \rangle = \sum_{i=1}^n \langle \varepsilon_i | \psi \rangle| \varepsilon_i \rangle = \left(\sum_{i=1}^n | \varepsilon_i \rangle \langle \varepsilon_i | \right ) | \psi \rangle \,$

we obtain the closure relation:

$\sum_{i=1}^n | \varepsilon_i \rangle \langle \varepsilon_i | = 1 .$

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:

$| \chi \rangle = \sum_{j = 1}^n z_j | \varepsilon_j \rangle = z_1 | \varepsilon_1 \rangle + z_2 | \varepsilon_2 \rangle + \cdots z_n | \varepsilon_n \rangle = \begin{bmatrix} \langle \varepsilon_1 | \chi \rangle \\ \vdots \\ \langle \varepsilon_n | \chi \rangle \end{bmatrix} = \begin{bmatrix} z_1 \\ \vdots \\ z_n \end{bmatrix} .$

then the inner product can be obtained:

$\langle \chi | \psi \rangle = \langle \chi | 1 | \psi \rangle = \langle \chi | \left( \sum_{i=1}^n | \varepsilon_i \rangle \langle \varepsilon_i | \right) | \psi \rangle = \sum_{i=1}^n \langle \chi | \varepsilon_i \rangle \langle \varepsilon_i | \psi \rangle = \sum_{i=1}^n z_i^{*} c_i.$

Normalization in discrete bases

The norm or magnitude of the state vector |ψ is:

$\|\psi\|^2 = \langle \psi | \psi \rangle = \sum_{j=1}^n | c_j |^2 .$

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:

$\frac{1}{\|\psi\|^2}\langle \psi | \psi \rangle = \frac{1}{\|\psi\|^2}\sum_{j=1}^n | c_j |^2 = 1 \,,$

so the normalized state |ψN in all generality is:

$| \psi_N \rangle = \frac{1}{\sqrt{\langle \psi|\psi\rangle}} | \psi \rangle$

Compare the similarity with Euclidean unit vectors a in elementary vector calculus:

$\mathbf{\hat{a}} = \frac{1}{\sqrt{\mathbf{a}\cdot\mathbf{a}}}\mathbf{a}$

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:

$\Psi \left ( s_z,t \right )$,

where again sz 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:

$\vec \psi = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}$

where c1 = Ψ+ and c2 = 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 |c1|2, and spin down (–1/2 ≡ ↓z) with probability |c2|2.

In bra–ket notation this can be written:

\begin{align} | \psi \rangle & = c_1 | \uparrow_z \rangle + c_2 | \downarrow_z \rangle \\ & = \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)

$| \uparrow_z \rangle \equiv | + \rangle$ for "spin up" or sz = +1/2,
$| \downarrow_z \rangle \equiv | - \rangle$ for "spin down" or sz = –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 c1) plus the probability in the spin down (z, corresponding to the coefficient c2) state is 1.

To see this explicitly for this case, expand the ket in terms of the bases:

$| \psi \rangle = c_1| \uparrow_z \rangle + c_2| \downarrow_z \rangle ,$

implying

$\langle \psi | = c_1^{*} \langle \uparrow_z | + c_2^{*} \langle \downarrow_z | ,$

taking the inner product (and recalling orthonormality) leads to the normalization condition:

\begin{align} \langle \psi | \psi \rangle & = \left ( c_1^{*} \langle \uparrow_z | + c_2^{*} \langle \downarrow_z | \right )\left ( c_1| \uparrow_z \rangle + c_2| \downarrow_z \rangle \right ) \\ & = \left ( c_1^{*} \langle \uparrow_z | + c_2^{*} \langle \downarrow_z | \right ) c_1| \uparrow_z \rangle + \left ( c_1^{*} \langle \uparrow_z | + c_2^{*} \langle \downarrow_z | \right ) c_2| \downarrow_z \rangle \\ & = 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;

$\left | \psi \right \rangle = \sum_{i = 1}^\infty c_i \left | \varepsilon_i \right \rangle = c_1 \left | \varepsilon_1 \right \rangle + c_2 \left | \varepsilon_2 \right \rangle + \cdots = \begin{bmatrix} \left \langle \varepsilon_1 | \psi \right \rangle \\ \vdots \\ \left \langle \varepsilon_n | \psi \right \rangle \\ \vdots \end{bmatrix} = \begin{bmatrix} c_1 \\ \vdots \\ c_n \\ \vdots \end{bmatrix} .$

with corresponding bra as before:

\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 |ψ 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 |ψ is given by:

$| \psi \rangle = \int d \varepsilon | \varepsilon \rangle \psi(\varepsilon) \,,$

with corresponding bra:

$\langle \psi | = \int d \varepsilon \langle \varepsilon | {\psi(\varepsilon)}^{*} \,,$

and again the basis here is orthonormal:

$\langle \varepsilon | \varepsilon' \rangle = \delta (\varepsilon-\varepsilon')\,.$

As with the discrete bases, a symbol ε is used in the basis states, two common notations are |ε and sometimes |ψε. A particular basis ket may be subscripted |ε0|ψε0 or primed |ε|ψε.

The components of the state |ψ are still ε|ψ, the projection of the state onto some basis is a function;

$\langle \varepsilon_0 | \psi \rangle = \langle \varepsilon_0 | \left( \int d \varepsilon | \varepsilon \rangle \psi(\varepsilon) \right) = \int d \varepsilon \langle \varepsilon_0 | \varepsilon \rangle \psi(\varepsilon) = \int d \varepsilon \delta( \varepsilon_0 - \varepsilon ) \psi(\varepsilon) = \psi(\varepsilon_0) \,,$

This time

$| \psi(\varepsilon) |^2 = | \langle \varepsilon | \psi \rangle |^2$

is the probability density function of measuring the observable ε, so integrating this with respect to ε between aεb gives:

$P_{a \leq \varepsilon \leq b} = \int_a^b d\varepsilon | \psi(\varepsilon) |^2 = \int_a^b d\varepsilon| \langle \varepsilon | \psi \rangle |^2 \,,$

the probability of finding the system with ε between ε = a and ε = b.

Closure relation in continuous bases

Taking the state above

$|\psi\rangle = \int d \varepsilon \, | \varepsilon \rangle \psi(\varepsilon) = \int d \varepsilon \, | \varepsilon \rangle \langle \varepsilon | \psi \rangle = \left(\int d\varepsilon \, | \varepsilon \rangle \langle \varepsilon | \right) | \psi \rangle \,$

we obtain the closure relation:

$\int d\varepsilon \, | \varepsilon \rangle \langle \varepsilon | = 1 \,$

Also the inner product can be obtained:

$\langle \chi | \psi \rangle = \langle \chi | 1 | \psi \rangle = \langle \chi| \left( \int d \varepsilon | \varepsilon \rangle \langle \varepsilon | \right) | \psi \rangle = \int d \varepsilon \langle \chi | \varepsilon \rangle \langle \varepsilon | \psi \rangle = \int d \varepsilon \chi(\varepsilon)^{*} \psi(\varepsilon) .$

Normalization in continuous bases

Taking the inner product;

$\langle \psi | \psi \rangle = \int d \varepsilon \, | \psi(\varepsilon) |^2 = \|\psi\|^2 .$

This integral is the total probability of all basis states occurring, so it must be 1 as before:

$\frac{1}{\|\psi\|^2}\langle \psi | \psi \rangle = \frac{1}{\|\psi\|^2} \int d \varepsilon | \psi(\varepsilon) |^2 = 1$

hence

$| \psi_N \rangle = \frac{1}{\sqrt{\langle \psi|\psi\rangle}} | \psi \rangle$

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 xpx in 1d or rp 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 |ψ can be expanded in terms of a continuum of states; i.e. |ε|ψε|x|ψx, corresponding to each x.

If the particle is confined to a region R (a subset of the x-axis), the state is:

$| \psi \rangle = \int\limits_R d x \, | x \rangle \langle x | \psi \rangle = \int\limits_R d x \, \psi(x) | x \rangle$

$1 = \int\limits_R d x \, | x \rangle \langle x |$

and the inner product as stated at the beginning of this article (in that case R = (−∞, ∞)):

$\langle \chi | \psi \rangle = \int\limits_R d x \, \langle \chi | x \rangle \langle x | \psi \rangle = \int\limits_R d x \, \chi(x)^{*} \psi(x) \,.$.

The "wavefunction" described previously is simply a component of the complex state vector. Projecting |ψ onto a particular position state |x0, where x0 is in R:

$\langle x_0 | \psi \rangle = \int\limits_R d x \, \langle x_0 | x \rangle \psi(x) = \int\limits_R d x \, \delta( x_0 - x ) \psi(x) = \psi(x_0) \,.$

One spin-0 particle in three dimensions

The generalization of the previous result is straightforward. In three dimensions, |ψ can be expanded in terms of a continuum of states with definite position, so |ε|ψε|r|x, y, z|ψr, corresponding to each r = (x, y, z).

If the particle is confined to a region R (a subset of 3d space), the state is;

$| \psi \rangle = \int\limits_R d^3\mathbf{r} \, | \mathbf{r} \rangle \langle \mathbf{r} | \psi\rangle = \int\limits_R d^3\mathbf{r} \, \psi(\mathbf{r}) | \mathbf{r} \rangle$

The closure relation is

$1 = \int\limits_R d^3\mathbf{r} \, | \mathbf{r} \rangle \langle \mathbf{r} |$

leading to the inner product of |ψ with itself leads to the normalization conditions in the three-dimensional definitions above:

$\langle \chi | \psi \rangle = \int\limits_R d^3\mathbf{r} \, \langle \chi | \mathbf{r} \rangle \langle \mathbf{r} | \psi \rangle = \int\limits_R d^3\mathbf{r} \, \chi(\mathbf{r})^{*} \psi(\mathbf{r})$.

Projecting $| \psi \rangle$ onto a particular position state |r0, where r0 is in R:

$\langle \mathbf{r}_0 | \psi \rangle = \int\limits_R d^3 \mathbf{r} \, \langle \mathbf{r}_0 | \mathbf{r} \rangle \psi(\mathbf{r}) = \int\limits_R d^3 \mathbf{r} \, \delta( \mathbf{r}_0 - \mathbf{r} ) \psi(\mathbf{r}) = \psi(\mathbf{r}_0)$

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 |r, sz are a combination of the discrete variable sz (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:

$| \Psi \rangle = \sum_{s_z} \int\limits_R d^3 \, \mathbf{r} \Psi(\mathbf{r},s_z) | \mathbf{r}, s_z \rangle$

and therefore the closure relation (identity operator) is:

$1 = \sum_{s_z} \int\limits_R d^3 \, \mathbf{r} | \mathbf{r},s_z\rangle \langle \mathbf{r} , s_z |$

Projecting Ψ onto a particular position-spin state |r0, m, where r0 is in R:

$\langle \mathbf{r}_0, m | \Psi \rangle = \sum_{s_z}\int\limits_R d^3 \mathbf{r} \, \langle \mathbf{r}_0, m | \mathbf{r}, s_z \rangle \Psi(\mathbf{r}, s_z) = \sum_{s_z}\int\limits_R d^3 \mathbf{r} \, \delta_{m \, s_z}\delta( \mathbf{r}_0 - \mathbf{r} ) \Psi(\mathbf{r}, s_z) = \Psi(\mathbf{r}_0, m) \,.$

where the joint orthogonality relation

$\langle \mathbf{r}_0, m | \mathbf{r}, s_z \rangle = \delta_{m\,s_z}\delta( \mathbf{r}_0 - \mathbf{r} )$

has been used.

Time dependence

In the Schrödinger picture, the states evolve in time, so the time dependence is placed in |ψ according to:[13]

$|\psi(t)\rangle = \sum_i \, | \varepsilon_i \rangle \langle \varepsilon_i | \psi(t)\rangle = \sum_i c_i(t) | \varepsilon \rangle$

for discrete bases, or

$|\psi(t)\rangle = \int d\varepsilon \, | \varepsilon \rangle \langle \varepsilon | \psi(t)\rangle = \int d\varepsilon \, \psi(\varepsilon,t) | \varepsilon \rangle$

for continuous bases. However, in the Heisenberg picture the states |ψ are constant in time and time dependence is placed in the Heisenberg operators, so |ψ is not written as |ψ(t).

Wave function collapse

The physical meaning of the components of |ψ 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 |εi has distinct and definite values, λi, and a measurement of that variable is performed on a system in the state |ψ then the probability of measuring λi is |εi|ψ|2. If the measurement yields λi, the system "collapses" to the state |εi, 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:

References

1. ^ R.G. Lerner, G.L. Trigg (1991). Encyclopaedia of Physics (2nd ed.). VHC Publishers. pp. 1223–1229. ISBN 0-89573-752-3.
2. ^ Hanle, P.A. (December 1977), "Erwin Schrodinger's Reaction to Louis de Broglie's Thesis on the Quantum Theory.", Isis 68 (4): 606–609, doi:10.1086/351880
3. ^ Physics for Scientists and Engineers – with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, ISBN 0-7167-8964-7
4. ^ Sears' and Zemansky's University Physics, Young and Freedman (12th edition), Pearson Ed. & Addison-Wesley Inc., 2008, ISBN 978-0-321-50130-1
5. ^ a b c Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, ISBN 0-19-855493-1
6. ^ McMahon, David (2008). Quantum Field Theory Demystified. McGraw Hill Professional. ISBN 9780071643528.
7. ^ Particle Physics (3rd Edition), B.R. Martin, G. Shaw, Manchester Physics Series, John Wiley & Sons, 2008, ISBN 978-0-470-03294-7
8. ^ a b c Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0
9. ^ Griffiths. Introduction to Quantum Mechanics (1st ed.). p. 107.
10. ^ Quantum Mechanics (3rd Edition), Eugen Merzbacher, 1998, John Wiley & Sons, ISBN 0-471-88702-1
11. ^ Griffiths, p179 of the first edition
12. ^ Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
13. ^ Y. Peleg, R. Pnini, E. Zaarur, E. Hecht (2010). Quantum mechanics. Schuam's outlines (2nd ed.). McGraw Hill. pp. 68–69. ISBN 978-0-07-162358-2.
14. ^ E. T. Jaynes. Probability Theory: The Logic of Science, Cambridge University Press (2003),
15. ^ Pusey, Matthew F.; Jonathan Barrett, Terry Rudolph (14). "The quantum state cannot be interpreted statistically". arXiv.org: arxiv:1111.3328v1.

2.Quantum Mechanics (Non-Relativistic Theory), L.D. Landau and E.M. Lifshitz, ISBN 0-08-020940-8