28 May 2025

Quantum Postulates, Wavefunctions, and Operators

quantum-postulates wavefunction probability-current hermitian-operators commutators

Postulates of quantum mechanics

1. A pure state is represented by a non-zero vector $\lvert\psi\rangle$ in a complex Hilbert space. Vectors differing only by a non-zero complex factor represent the same physical ray. In the position representation, $\psi(\mathbf r,t)=\langle\mathbf r\vert\psi(t)\rangle$.
2. Each observable $A$ is represented by a linear Hermitian operator $\hat A$. Its possible measured values are its eigenvalues, defined by $\hat A\phi_a=a\phi_a$.
3. If a normalized state is expanded in an orthonormal eigenbasis as $\lvert\psi\rangle=\sum_a c_a\lvert a\rangle$, a measurement of $A$ gives $a$ with probability $\lvert c_a\rvert^2$. Immediately after an ideal measurement with result $a$, the state lies in the corresponding eigenspace.
4. The expectation value in a normalized state is $\langle A\rangle=\langle\psi\vert\hat A\vert\psi\rangle$.
5. Between measurements the state evolves according to $i\hbar\,\partial_t\lvert\psi\rangle=\hat H\lvert\psi\rangle$, where $\hat H$ is the Hamiltonian.

Probability interpretation and normalization

Born’s interpretation identifies

\[\boxed{\rho(\mathbf r,t)=\lvert\psi(\mathbf r,t)\rvert^2}\]

as probability per unit volume. Thus $\rho$ has SI units $\mathrm{m^{-3}}$, and the probability of finding the particle in a volume $\Omega$ is

\[P(\Omega,t)=\int_\Omega \lvert\psi(\mathbf r,t)\rvert^2d^3r.\]

A state representing one particle over all space is normalized by

\[\boxed{\int_{\mathbb R^3}\lvert\psi\rvert^2d^3r=1}.\]

If an unnormalized square-integrable function is $f$, then $\psi=Nf$ with

\[\lvert N\rvert^{-2}=\int_{\mathbb R^3}\lvert f\rvert^2d^3r.\]

The overall phase of $N$ is physically irrelevant because it cancels from $\lvert\psi\rvert^2$ and all expectation values.

Physical acceptability of a wavefunction

A bound-state wavefunction must be single-valued, finite, and square-integrable. It is continuous wherever the potential has no infinite discontinuity. To obtain the derivative condition in one dimension, integrate the stationary Schrödinger equation across $(x_0-\epsilon,x_0+\epsilon)$:

\[-\frac{\hbar^2}{2m}\int_{x_0-\epsilon}^{x_0+\epsilon}\psi''dx +\int_{x_0-\epsilon}^{x_0+\epsilon}(V-E)\psi\,dx=0.\]

Therefore

\[\psi'(x_0+\epsilon)-\psi'(x_0-\epsilon) =\frac{2m}{\hbar^2}\int_{x_0-\epsilon}^{x_0+\epsilon}(V-E)\psi\,dx.\]

If $V$ and $\psi$ remain finite, the integral vanishes as $\epsilon\to0$, so $\psi’$ is continuous. At an infinite wall this argument fails and the physical boundary condition is instead $\psi=0$ at the wall; its derivative need not match the identically zero exterior derivative. For a normalizable state on an unbounded domain, $\psi$ must approach zero sufficiently rapidly as $\lvert\mathbf r\rvert\to\infty$.

Linearity and superposition

The Schrödinger equation is linear. If $\psi_1$ and $\psi_2$ satisfy the same homogeneous equation, then for constants $c_1,c_2$,

\[(i\hbar\partial_t-\hat H)(c_1\psi_1+c_2\psi_2) =c_1(i\hbar\partial_t-\hat H)\psi_1 +c_2(i\hbar\partial_t-\hat H)\psi_2=0.\]

Hence $c_1\psi_1+c_2\psi_2$ is also a state after normalization. The probability density contains cross terms $c_1c_2^{\ast}\psi_1\psi_2^{\ast}+\mathrm{c.c.}$; these interference terms distinguish a coherent superposition from a classical addition of probabilities.

Probability current density in three dimensions

For a real potential,

\[i\hbar\frac{\partial\psi}{\partial t} =-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi,\]

and complex conjugation reverses the sign of $i$:

\[-i\hbar\frac{\partial\psi^{\ast}}{\partial t} =-\frac{\hbar^2}{2m}\nabla^2\psi^{\ast}+V\psi^{\ast}.\]

Multiply the first equation by $\psi^{\ast}$, the second by $\psi$, and subtract the second from the first. The potential terms cancel:

\[i\hbar\frac{\partial\lvert\psi\rvert^2}{\partial t} =-\frac{\hbar^2}{2m} (\psi^{\ast}\nabla^2\psi-\psi\nabla^2\psi^{\ast}).\]

Using

\[\psi^{\ast}\nabla^2\psi-\psi\nabla^2\psi^{\ast} =\nabla\!\cdot(\psi^{\ast}\nabla\psi-\psi\nabla\psi^{\ast}),\]

we obtain the continuity equation

\[\boxed{\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf j=0},\]

with

\[\boxed{ \mathbf j=\frac{\hbar}{2mi} (\psi^{\ast}\nabla\psi-\psi\nabla\psi^{\ast}) =\frac{\hbar}{m}\operatorname{Im}(\psi^{\ast}\nabla\psi) }.\]

Since $\rho$ has units $\mathrm{m^{-3}}$, $\mathbf j$ has units $\mathrm{m^{-2}s^{-1}}$. Integrating over a fixed volume and applying the divergence theorem gives

\[\frac d{dt}\int_\Omega\rho\,d^3r =-\oint_{\partial\Omega}\mathbf j\cdot d\mathbf S,\]

so probability decreases inside $\Omega$ only by flowing through its boundary.

Free-particle wavefunction

For $V=0$, a simultaneous momentum and energy eigenfunction is

\[\psi_{\mathbf p}(\mathbf r,t)=A \exp\!\left[\frac{i}{\hbar}(\mathbf p\cdot\mathbf r-Et)\right], \qquad E=\frac{p^2}{2m}.\]

Substitution gives $\nabla^2\psi=-p^2\psi/\hbar^2$ and $\partial_t\psi=-iE\psi/\hbar$, fixing both signs in the Schrödinger equation. Because $\lvert\psi_{\mathbf p}\rvert^2=\lvert A\rvert^2$ is constant, a plane wave cannot be normalized to unity over infinite space. The standard generalized normalization is

\[\langle\mathbf r\vert\mathbf p\rangle =\frac{1}{(2\pi\hbar)^{3/2}} e^{i\mathbf p\cdot\mathbf r/\hbar},\] \[\langle\mathbf p\vert\mathbf p'\rangle =\delta^{(3)}(\mathbf p-\mathbf p').\]

A physical localized free state is a square-integrable superposition of these plane waves.

Eigenfunctions, eigenvalues, and Hermitian operators

An eigenfunction satisfies

\[\hat A\phi_a=a\phi_a.\]

Hermiticity means

\[\langle\phi\vert\hat A\psi\rangle =\langle\hat A\phi\vert\psi\rangle\]

for functions obeying the operator’s boundary conditions. If $\hat A\phi_a=a\phi_a$, then

\[a\langle\phi_a\vert\phi_a\rangle =\langle\phi_a\vert\hat A\phi_a\rangle =\langle\hat A\phi_a\vert\phi_a\rangle =a^{\ast}\langle\phi_a\vert\phi_a\rangle,\]

so $a=a^{\ast}$: an observable eigenvalue is real. For two eigenfunctions with $a\neq b$,

\[b\langle\phi_a\vert\phi_b\rangle =a\langle\phi_a\vert\phi_b\rangle,\]

which implies $\langle\phi_a\vert\phi_b\rangle=0$. Distinct eigenvalues of a Hermitian operator therefore have orthogonal eigenfunctions.

Position, momentum, and energy operators

Acting on the plane-wave phase gives

\[-i\hbar\nabla e^{i(\mathbf p\cdot\mathbf r-Et)/\hbar} =\mathbf p\,e^{i(\mathbf p\cdot\mathbf r-Et)/\hbar},\] \[i\hbar\partial_t e^{i(\mathbf p\cdot\mathbf r-Et)/\hbar} =E\,e^{i(\mathbf p\cdot\mathbf r-Et)/\hbar}.\]

Thus the position representation uses

\[\boxed{\hat{\mathbf r}=\mathbf r},\qquad \boxed{\hat{\mathbf p}=-i\hbar\nabla},\qquad \boxed{\hat E=i\hbar\frac{\partial}{\partial t}}.\]

For one Cartesian component and any differentiable test function $f$,

\[\begin{aligned} [\hat x,\hat p_x]f &=x(-i\hbar f')-(-i\hbar)(xf)'\\ &=-i\hbar xf'+i\hbar(f+xf')=i\hbar f. \end{aligned}\]

Hence

\[\boxed{[\hat x,\hat p_x]=i\hbar}.\]

Position and momentum are non-commuting operators and cannot have a complete common eigenbasis. Commuting Hermitian operators, $[\hat A,\hat B]=0$, can be simultaneously diagonalized after resolving any degeneracy.

Finally, for a normalized wavefunction,

\[\boxed{\langle\mathbf r\rangle =\int\psi^{\ast}\mathbf r\psi\,d^3r},\] \[\boxed{\langle\mathbf p\rangle =\int\psi^{\ast}(-i\hbar\nabla)\psi\,d^3r}.\]

Integration by parts shows that $-i\hbar\nabla$ is Hermitian when the surface term vanishes under the physical boundary conditions.