Following are the general forms of the wave function for systems in higher dimensions and more particles, as well as including other degrees of freedom than position coordinates or momentum components.
Finite dimensional Hilbert space While
Hilbert spaces originally refer to infinite dimensional
complete inner product spaces they, by definition, include finite dimensional
complete inner product spaces as well. In physics, they are often referred to as
finite dimensional Hilbert spaces. For every finite dimensional Hilbert space there exist
orthonormal basis kets that
span the entire Hilbert space. If the -dimensional set \{ |\phi_i\rangle \} is orthonormal, then the projection operator for the space spanned by these states is given by: P = \sum_i |\phi_i\rangle\langle \phi_i | = I where the projection is equivalent to identity operator since \{ |\phi_i\rangle \} spans the entire Hilbert space, thus leaving any vector from Hilbert space unchanged. This is also known as completeness relation of finite dimensional Hilbert space. The wavefunction is instead given by: |\psi\rangle = I|\psi\rangle = \sum_i |\phi_i\rangle\langle \phi_i |\psi\rangle where \{ \langle \phi_i |\psi\rangle \} , is a set of complex numbers which can be used to construct a wavefunction using the above formula.
Probability interpretation of inner product If the set \{ |\phi_i\rangle \} are eigenkets of a non-
degenerate observable with eigenvalues \lambda_i, by the
postulates of quantum mechanics, the probability of measuring the observable to be \lambda_i is given according to
Born rule as: P_\psi(\lambda_i) = |\langle \phi_i|\psi \rangle|^2 For non-degenerate \{ |\phi_i\rangle \} of some observable, if eigenvalues \lambda have subset of eigenvectors labelled as \{ |\lambda^{(j)}\rangle \}, by the
postulates of quantum mechanics, the probability of measuring the observable to be \lambda is given by: P_\psi(\lambda) =\sum_j |\langle \lambda^{(j)}|\psi \rangle|^2 = |\widehat P_\lambda |\psi \rangle |^2 where \widehat P_\lambda =\sum_j|\lambda^{(j)}\rangle\langle\lambda^{(j)}| is a projection operator of states to subspace spanned by \{ |\lambda^{(j)}\rangle \}. The equality follows due to orthogonal nature of \{ |\phi_i\rangle \}. Hence, \{ \langle \phi_i |\psi\rangle \} which specify state of the quantum mechanical system, have magnitudes whose square gives the probability of measuring the respective |\phi_i\rangle state.
Physical significance of relative phase While the relative phase has observable effects in experiments, the global phase of the system is experimentally indistinguishable. For example in a particle in superposition of two states, the global phase of the particle cannot be distinguished by finding expectation value of observable or probabilities of observing different states but relative phases can affect the expectation values of observables. While the overall phase of the system is considered to be arbitrary, the relative phase for each state |\phi_i\rangle of a prepared state in superposition can be determined based on physical meaning of the prepared state and its symmetry. For example, the construction of spin states along x direction as a superposition of spin states along z direction, can done by applying appropriate rotation transformation on the spin along z states which provides appropriate phase of the states relative to each other.
Application to include spin An example of finite dimensional Hilbert space can be constructed using spin eigenkets of s-spin particles which forms a 2s+1 dimensional
Hilbert space. However, the general wavefunction of a particle that fully describes its state, is always from an infinite dimensional
Hilbert space since it involves a tensor product with
Hilbert space relating to the position or momentum of the particle. Nonetheless, the techniques developed for finite dimensional Hilbert space are useful since they can either be treated independently or treated in consideration of linearity of tensor product. Since the
spin operator for a given s-spin particles can be represented as a finite (2s+1)^2
matrix which acts on 2s+1 independent spin vector components, it is usually preferable to denote spin components using matrix/column/row notation as applicable. For example, each is usually identified as a column vector:|s\rangle \leftrightarrow \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \\ 0 \\ \end{bmatrix} \,, \quad |s-1\rangle \leftrightarrow \begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \\ 0 \\ \end{bmatrix} \,, \ldots \,, \quad |-(s-1)\rangle \leftrightarrow \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \\ 0 \\ \end{bmatrix} \,,\quad |-s\rangle \leftrightarrow \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \\ \end{bmatrix} but it is a common abuse of notation, because the kets are not synonymous or equal to the column vectors. Column vectors simply provide a convenient way to express the spin components. Corresponding to the notation, the z-component spin operator can be written as:\frac{1}{\hbar}\hat{S}_z = \begin{bmatrix} s & 0 & \cdots & 0 & 0 \\ 0 & s-1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & -(s-1) & 0 \\ 0 & 0 & \cdots & 0 & -s \end{bmatrix} since the
eigenvectors of z-component spin operator are the above column vectors, with eigenvalues being the corresponding spin quantum numbers. Corresponding to the notation, a vector from such a finite dimensional Hilbert space is hence represented as: |\phi\rangle = \begin{bmatrix} \langle s| \phi\rangle \\ \langle s-1| \phi\rangle \\ \vdots \\ \langle -(s-1)| \phi\rangle \\ \langle -s| \phi\rangle \\ \end{bmatrix} =\begin{bmatrix} \varepsilon_s \\ \varepsilon_{s-1}\\ \vdots \\ \varepsilon_{-s+1} \\ \varepsilon_{-s} \\ \end{bmatrix} where \{ \varepsilon_i \} are corresponding complex numbers. In the following discussion involving spin, the complete wavefunction is considered as tensor product of spin states from finite dimensional Hilbert spaces and the wavefunction which was previously developed. The basis for this Hilbert space are hence considered: |\mathbf{r}, s_z\rangle = |\mathbf{r}\rangle |s_z\rangle .
One-particle states in 3d position space The position-space wave function of a single particle without spin in three spatial dimensions is similar to the case of one spatial dimension above: \Psi(\mathbf{r},t) where is the
position vector in three-dimensional space, and is time. As always is a complex-valued function of real variables. As a single vector in
Dirac notation |\Psi(t)\rangle = \int d^3\! \mathbf{r}\, \Psi(\mathbf{r},t) \,|\mathbf{r}\rangle All the previous remarks on inner products, momentum space wave functions, Fourier transforms, and so on extend to higher dimensions. For a particle with
spin, ignoring the position degrees of freedom, the wave function is a function of spin only (time is a parameter); \xi(s_z,t) where is the
spin projection quantum number along the axis. (The axis is an arbitrary choice; other axes can be used instead if the wave function is transformed appropriately, see below.) The parameter, unlike and , is a
discrete variable. For example, for a
spin-1/2 particle, can only be or , and not any other value. (In general, for spin , can be ). Inserting each quantum number gives a complex valued function of space and time, there are of them. These can be arranged into a
column vector \xi = \begin{bmatrix} \xi(s,t) \\ \xi(s-1,t) \\ \vdots \\ \xi(-(s-1),t) \\ \xi(-s,t) \\ \end{bmatrix} = \xi(s,t) \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \\ 0 \\ \end{bmatrix} + \xi(s-1,t)\begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \\ 0 \\ \end{bmatrix} + \cdots + \xi(-(s-1),t) \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \\ 0 \\ \end{bmatrix} + \xi(-s,t) \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \\ \end{bmatrix} In
bra–ket notation, these easily arrange into the components of a vector: |\xi (t)\rangle = \sum_{s_z=-s}^s \xi(s_z,t) \,| s_z \rangle The entire vector is a solution of the Schrödinger equation (with a suitable Hamiltonian), which unfolds to a coupled system of ordinary differential equations with solutions . The term "spin function" instead of "wave function" is used by some authors. This contrasts the solutions to position space wave functions, the position coordinates being continuous degrees of freedom, because then the Schrödinger equation does take the form of a wave equation. More generally, for a particle in 3d with any spin, the wave function can be written in "position–spin space" as: \Psi(\mathbf{r},s_z,t) and these can also be arranged into a column vector \Psi(\mathbf{r},t) = \begin{bmatrix} \Psi(\mathbf{r},s,t) \\ \Psi(\mathbf{r},s-1,t) \\ \vdots \\ \Psi(\mathbf{r},-(s-1),t) \\ \Psi(\mathbf{r},-s,t) \\ \end{bmatrix} in which the spin dependence is placed in indexing the entries, and the wave function is a complex
vector-valued function of space and time only. All values of the wave function, not only for discrete but
continuous variables also, collect into a single vector |\Psi(t)\rangle = \sum_{s_z}\int d^3\!\mathbf{r} \,\Psi(\mathbf{r},s_z,t)\, |\mathbf{r}, s_z\rangle For a single particle, the
tensor product of its position state vector and spin state vector gives the composite position-spin state vector |\psi(t)\rangle\! \otimes\! |\xi(t)\rangle = \sum_{s_z}\int d^3\! \mathbf{r}\, \psi(\mathbf{r},t)\,\xi(s_z,t) \,|\mathbf{r}\rangle \!\otimes\! |s_z\rangle with the identifications |\Psi (t)\rangle = |\psi(t)\rangle \!\otimes\! |\xi(t)\rangle \Psi(\mathbf{r},s_z,t) = \psi(\mathbf{r},t)\,\xi(s_z,t) |\mathbf{r},s_z \rangle= |\mathbf{r}\rangle \!\otimes\! |s_z\rangle The tensor product factorization of energy eigenstates is always possible if the orbital and spin angular momenta of the particle are separable in the
Hamiltonian operator underlying the system's dynamics (in other words, the Hamiltonian can be split into the sum of orbital and spin terms). The time dependence can be placed in either factor, and time evolution of each can be studied separately. Under such Hamiltonians, any tensor product state evolves into another tensor product state, which essentially means any unentangled state remains unentangled under time evolution. This is said to happen when there is no physical interaction between the states of the tensor products. In the case of non separable Hamiltonians, energy eigenstates are said to be some linear combination of such states, which need not be factorizable; examples include a particle in a
magnetic field, and
spin–orbit coupling. The preceding discussion is not limited to spin as a discrete variable, the total
angular momentum J may also be used. Other discrete degrees of freedom, like
isospin, can expressed similarly to the case of spin above.
Many-particle states in 3d position space If 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 particles is written: \Psi(\mathbf{r}_1,\mathbf{r}_2 \cdots \mathbf{r}_N,t) where is the position of the -th particle in three-dimensional space, and is time. Altogether, this is a complex-valued function of real variables. In quantum mechanics there is a fundamental distinction between
identical particles and
distinguishable particles. For example, any two electrons are identical and 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. This translates to a requirement on the wave function for a system of identical particles: \Psi \left ( \ldots \mathbf{r}_a, \ldots , \mathbf{r}_b, \ldots \right ) = \pm \Psi \left ( \ldots \mathbf{r}_b, \ldots , \mathbf{r}_a, \ldots \right ) where the sign occurs if the particles are
all bosons and sign if they are
all fermions. In other words, the wave function is either totally symmetric in the positions of bosons, or totally antisymmetric in the positions of fermions. The physical interchange of particles corresponds to mathematically switching arguments in the wave function. The antisymmetry feature of fermionic wave functions leads to the
Pauli principle. Generally, bosonic and fermionic symmetry requirements are the manifestation of
particle statistics and are present in other quantum state formalisms. For
distinguishable particles (no two being
identical, i.e. no two having the same set of quantum numbers), there is no requirement for the wave function to be either symmetric or antisymmetric. For a collection of particles, some identical with coordinates and others distinguishable (not identical with each other, and not identical to the aforementioned identical particles), the wave function is symmetric or antisymmetric in the identical particle coordinates only: \Psi \left ( \ldots \mathbf{r}_a, \ldots , \mathbf{r}_b, \ldots , \mathbf{x}_1, \mathbf{x}_2, \ldots \right ) = \pm \Psi \left ( \ldots \mathbf{r}_b, \ldots , \mathbf{r}_a, \ldots , \mathbf{x}_1, \mathbf{x}_2, \ldots \right ) Again, there is no symmetry requirement for the distinguishable particle coordinates . The wave function for
N particles each with spin is the complex-valued function \Psi(\mathbf{r}_1, \mathbf{r}_2 \cdots \mathbf{r}_N, s_{z\,1}, s_{z\,2} \cdots s_{z\,N}, t) Accumulating all these components into a single vector, | \Psi \rangle = \overbrace{\sum_{s_{z\,1},\ldots,s_{z\,N}}}^{\text{discrete labels}} \overbrace{\int_{R_N} d^3\mathbf{r}_N \cdots \int_{R_1} d^3\mathbf{r}_1}^{\text{continuous labels}} \; \underbrace{{\Psi}( \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} )}_{\begin{array}{c}\text{wave function (component of } \\ \text{ state vector along basis state)}\end{array}} \; \underbrace{| \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} \rangle }_{\text{basis state (basis ket)}}\,. For identical particles, symmetry requirements apply to both position and spin arguments of the wave function so it has the overall correct symmetry. The formulae for the inner products are integrals over all coordinates or momenta and sums over all spin quantum numbers. For the general case of particles with spin in 3-d, ( \Psi_1 , \Psi_2 ) = \sum_{s_{z\,N}} \cdots \sum_{s_{z\,2}} \sum_{s_{z\,1}} \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_1 \int\limits_{\mathrm{ all \, space}} d ^3\mathbf{r}_2\cdots \int\limits_{\mathrm{ all \, space}} d ^3 \mathbf{r}_N \Psi^{*}_1 \left(\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right )\Psi_2 \left(\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) this is altogether three-dimensional
volume integrals and sums over the spins. The differential volume elements are also written "" or "". The multidimensional Fourier transforms of the position or position–spin space wave functions yields momentum or momentum–spin space wave functions.
Probability interpretation For the general case of particles with spin in 3d, if is interpreted as a probability amplitude, the probability density is \rho\left(\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) = \left | \Psi\left (\mathbf{r}_1 \cdots \mathbf{r}_N,s_{z\,1}\cdots s_{z\,N},t \right ) \right |^2 and the probability that particle 1 is in region with spin
and particle 2 is in region with spin etc. at time is the integral of the probability density over these regions and evaluated at these spin numbers: :P_{\mathbf{r}_1\in R_1,s_{z\,1} = m_1, \ldots, \mathbf{r}_N\in R_N,s_{z\,N} = m_N} (t) = \int_{R_1} d ^3\mathbf{r}_1 \int_{R_2} d ^3\mathbf{r}_2\cdots \int_{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
Physical significance of phase In non-relativistic quantum mechanics, it can be shown using Schrodinger's time dependent wave equation that the equation: \frac{\partial \rho}{\partial t} + \nabla\cdot\mathbf J = 0 is satisfied, where \rho(\mathbf x,t) = | \psi(\mathbf x,t)|^2 is the probability density and \mathbf J(\mathbf x,t) = \frac{\hbar}{2im}(\psi^* \nabla\psi-\psi\nabla\psi^*) = \frac{\hbar}{m} \text{Im}(\psi^* \nabla\psi) , is known as the
probability flux in accordance with the continuity equation form of the above equation. Using the following expression for wavefunction:\psi(\mathbf x,t)= \sqrt{\rho(\mathbf x,t)}\exp{\frac{iS(\mathbf x,t )}{\hbar}} where \rho(\mathbf x,t) = | \psi(\mathbf x,t)|^2 is the probability density and S(\mathbf x,t) is the phase of the wavefunction, it can be shown that: \mathbf J(\mathbf x,t) = \frac{\rho \nabla S}{m} Hence the spacial variation of phase characterizes the
probability flux. In classical analogy, for \mathbf J = \rho \mathbf v , the quantity \frac{\nabla S}{m} is analogous with velocity. Note that this does not imply a literal interpretation of \frac{\nabla S}{m} as velocity since velocity and position cannot be simultaneously determined as per the
uncertainty principle. Substituting the form of wavefunction in Schrodinger's time dependent wave equation, and taking the classical limit, \hbar |\nabla^2 S| \ll |\nabla S|^2 : \frac{1}{2m} |\nabla S(\mathbf x, t)|^2 + V(\mathbf x) + \frac{\partial S}{\partial t} = 0 Which is analogous to
Hamilton-Jacobi equation from classical mechanics. This interpretation fits with
Hamilton–Jacobi theory, in which \mathbf{P}_{\text{class.}} = \nabla S , where '''' is
Hamilton's principal function. == Time dependence ==