Wave-Mechanics Postulates

Wave mechanics is the position-representation form of nonrelativistic quantum mechanics. A pure state is represented by a wavefunction, observables act as multiplication or differential operators, position probabilities are integrals of a probability density, and closed-system dynamics is governed by the Schrödinger equation together with an operator domain and boundary conditions.

This representation is adapted to particles moving through space, so it is the practical language of wells, barriers, tunneling, wave packets, atoms, and scattering. It is not a theory separate from the Minimal Postulates. A wavefunction is a coordinate representation of a state, just as a column is a basis representation of a finite-dimensional state.

The Package at a Glance

For one spinless particle in a configuration region $\Omega\subseteq\mathbb R^d$:

1. State: a pure state is a ray represented by $\psi\in L^2(\Omega,d\mu)$.

2. Normalization: choose a representative satisfying $\int_\Omega\lvert\psi\rvert^2d\mu=1$.

3. Observables: suitable self-adjoint operators act on wavefunctions, often through multiplication or differentiation on declared domains.

4. Position Born rule: for a measurable region $R\subseteq\Omega$,

   $$\Pr(\mathbf r\in R) = \int_R \lvert\psi(\mathbf r)\rvert^2 d\mu(\mathbf r).$$

5. Dynamics: a closed-system wavefunction satisfies

   $$i\hbar \frac{\partial\psi}{\partial t} = \hat H\psi.$$

6. Boundary conditions: the Hilbert space, operator domain, and boundary conditions complete the physical specification of differential operators.

7. Composition: a multiparticle wavefunction lives on configuration space; product functions represent product states, while nonfactorizable functions represent entanglement.

The first six items are the traditional wave-mechanics package. The seventh makes explicit how the abstract composition postulate appears in coordinates.

Scope

The simplest setting is a single spinless particle with Cartesian coordinate $\mathbf r\in\mathbb R^3$. Its position representation is

$$\mathcal H \cong L^2(\mathbb R^3,d^3r).$$

If the particle is confined to a region $\Omega$, one instead uses an appropriate $L^2(\Omega,d\mu)$ space and an operator domain encoding the boundary conditions. The measure $d\mu$ is part of the representation.

This page assumes:

• a nonrelativistic particle with an external time parameter;
• a scalar wavefunction unless spin or internal components are stated;
• standard complex Hilbert-space quantum mechanics;
• sufficiently regular potentials for the displayed differential formulas;
• normalized states when probabilities are quoted.

Spin introduces multicomponent wavefunctions, mixed states require density operators or integral kernels, identical particles require permutation symmetry, and variable particle number leads toward Fock space. Relativistic quantum field theory is not obtained by merely replacing the Schrödinger equation with a relativistic-looking one-particle equation.

Postulate 1: The Wavefunction State

Let $\lvert\psi\rangle$ be an abstract pure state. Its position-space wavefunction is the amplitude

$$\psi(\mathbf r) = \langle\mathbf r\mid\psi\rangle.$$

The symbols $\lvert\mathbf r\rangle$ are generalized position eigenkets, not normalizable vectors in $L^2$. They are distributional objects used to express the position representation. The physical state is the normalizable vector reconstructed from its amplitudes, not an exact-position ket.

Rays and global phase

Multiplying every amplitude by one constant phase does not change the pure state:

$$\psi(\mathbf r) \sim e^{i\alpha}\psi(\mathbf r).$$

Relative phase does matter. If

$$\psi = c_1\psi_1+c_2\psi_2,$$

then

$$\begin{aligned} \lvert\psi\rvert^2 &= \lvert c_1\psi_1\rvert^2 + \lvert c_2\psi_2\rvert^2\\ &\quad+ 2\operatorname{Re} \left( c_1c_2^*\psi_1\psi_2^* \right). \end{aligned}$$

The last term is the interference term. A common overall phase cancels, whereas changing the phase between the two components changes this term.

Equality almost everywhere

An element of $L^2(\Omega,d\mu)$ is technically an equivalence class of functions. Two representatives define the same Hilbert-space vector if they differ only on a set of measure zero. Consequently, assigning a different value to a wavefunction at one isolated point does not create a new state or change a position probability.

This fact does not make pointwise regularity irrelevant. Differential operators require domains whose representatives possess suitable differentiability and boundary behavior. The Hilbert space tells us which states have finite norm; the operator domain tells us which of those states an unbounded operator may act on.

Mixed states

Not every state is represented by one wavefunction. A density operator can be represented by an integral kernel

$$\rho(\mathbf r,\mathbf r') = \langle\mathbf r\rvert \hat\rho \lvert\mathbf r'\rangle.$$

For a pure state,

$$\rho_\psi(\mathbf r,\mathbf r') = \psi(\mathbf r)\psi(\mathbf r')^*.$$

The diagonal gives the position density,

$$\rho_\psi(\mathbf r,\mathbf r) = \lvert\psi(\mathbf r)\rvert^2,$$

but off-diagonal entries carry coherence information. The Density-Matrix Formulation is the canonical general-state presentation.

Normalization, Inner Products, and Measure

A normalized scalar wavefunction obeys

$$\int_\Omega \lvert\psi(\mathbf r)\rvert^2 d\mu(\mathbf r) =1.$$

The inner product is

$$\langle\phi\mid\psi\rangle = \int_\Omega \phi(\mathbf r)^* \psi(\mathbf r) d\mu(\mathbf r).$$

Orthogonality means this integral vanishes. Completeness of an orthonormal discrete basis $\{\varphi_n\}$ means that a suitable state can be expanded as

$$\psi(\mathbf r) = \sum_n c_n\varphi_n(\mathbf r),$$

with

$$c_n = \int_\Omega \varphi_n(\mathbf r)^* \psi(\mathbf r) d\mu(\mathbf r).$$

Parseval’s identity then gives

$$\sum_n\lvert c_n\rvert^2 = \int_\Omega \lvert\psi(\mathbf r)\rvert^2 d\mu(\mathbf r).$$

Coordinate measures

In Cartesian coordinates,

$$d\mu(\mathbf r) = d^3r = dx\,dy\,dz.$$

In spherical coordinates,

$$d^3r = r^2\sin\theta \,dr\,d\theta\,d\phi.$$

More generally, a coordinate change $\mathbf r=\mathbf r(\mathbf q)$ introduces the Jacobian:

$$d^dr = \left\lvert \det\frac{\partial\mathbf r}{\partial\mathbf q} \right\rvert d^dq.$$

The wavefunction and measure must be interpreted together. Omitting the Jacobian changes normalization, expectation values, and probabilities.

Physical dimensions

Because a probability is dimensionless, a Cartesian $d$-dimensional wavefunction has units

$$[\psi] = \mathsf L^{-d/2}.$$

Thus $\lvert\psi(\mathbf r)\rvert^2$ is a density with units $\mathsf L^{-d}$, not a dimensionless probability. A probability appears only after integration over a region.

The site-wide normalization convention is summarized in Wavefunction Normalization.

Postulate 2: Operators in Position Representation

An abstract operator becomes a rule acting on wavefunctions. In Cartesian position representation,

$$(\hat{\mathbf r}\psi)(\mathbf r) = \mathbf r\,\psi(\mathbf r),$$

while momentum acts formally as

$$(\hat{\mathbf p}\psi)(\mathbf r) = -i\hbar\nabla\psi(\mathbf r).$$

For a spinless particle in a scalar potential,

$$\hat H = \frac{\hat{\mathbf p}^{\,2}}{2m} + V(\hat{\mathbf r},t)$$

becomes

$$(\hat H\psi)(\mathbf r) = -\frac{\hbar^2}{2m} \nabla^2\psi(\mathbf r) + V(\mathbf r,t)\psi(\mathbf r).$$

An expectation value is

$$\langle A\rangle_\psi = \int_\Omega \psi(\mathbf r)^* (\hat A\psi)(\mathbf r) d\mu(\mathbf r),$$

provided $\psi$ belongs to the required domain and the integral exists.

Differential expression versus operator

The symbols $-i\hbar\,d/dx$ and $-\hbar^2d^2/(2m\,dx^2)$ are differential expressions. A physical operator also requires:

1. a Hilbert space;
2. a domain of functions on which it acts;
3. boundary or asymptotic conditions;
4. a specification of how singular points are handled.

The same differential expression can define different operators with different spectra. On a line, a box, a ring, and a half-line, the kinetic-energy expression is accompanied by different spaces and domains.

In finite dimensions, Hermitian matrices are automatically self-adjoint. For unbounded differential operators, symmetry of matrix elements on a convenient set is not enough; the operator and its adjoint must have the same domain. See Hermitian vs Self-Adjoint Operators for the canonical distinction.

Postulate 3: The Born Rule in Position Space

For a normalized pure state, the probability of detecting the particle in a measurable region $R\subseteq\Omega$ is

$$\Pr(\mathbf r\in R) = \int_R \lvert\psi(\mathbf r)\rvert^2 d\mu(\mathbf r).$$

The associated spectral projector is formally

$$\hat P_R = \int_R \lvert\mathbf r\rangle \langle\mathbf r\rvert d\mu(\mathbf r),$$

so the abstract Born rule gives

$$\langle\psi\rvert \hat P_R \lvert\psi\rangle = \int_R \lvert\psi(\mathbf r)\rvert^2 d\mu(\mathbf r).$$

The density at a point is not the probability of that exact point. For a continuous distribution,

$$\Pr(\mathbf r=\mathbf r_0) =0$$

under ordinary conditions, while

$$\Pr(\mathbf r\in d^dr) \approx \lvert\psi(\mathbf r)\rvert^2d^dr$$

describes a sufficiently small detector cell. Real detectors have finite spatial resolution and are more accurately represented by effects associated with finite regions or response functions.

Other observables

The position-space wavefunction does not privilege position as the only observable. If $\hat A$ has a normalized discrete eigenfunction $\varphi_a(\mathbf r)$, then

$$\Pr(a) = \left\lvert \int_\Omega \varphi_a(\mathbf r)^* \psi(\mathbf r) d\mu(\mathbf r) \right\rvert^2.$$

For momentum in $d$ Cartesian dimensions, use the Fourier convention

$$\widetilde\psi(\mathbf p) = \frac{1} {(2\pi\hbar)^{d/2}} \int_{\mathbb R^d} e^{-i\mathbf p\cdot\mathbf r/\hbar} \psi(\mathbf r) d^dr.$$

Then

$$\int_{\mathbb R^d} \lvert\widetilde\psi(\mathbf p)\rvert^2 d^dp =1,$$

and momentum probabilities are obtained from $\lvert\widetilde\psi(\mathbf p)\rvert^2$. The functions $\psi(\mathbf r)$ and $\widetilde\psi(\mathbf p)$ represent the same state in different generalized bases. See Momentum Representation and Born Rule for Continuous Spectra.

Postulate 4: The Schrödinger Equation

For a closed system, the abstract equation

$$i\hbar \frac{d}{dt} \lvert\psi(t)\rangle = \hat H(t)\lvert\psi(t)\rangle$$

becomes

$$i\hbar \frac{\partial}{\partial t} \psi(\mathbf r,t) = \hat H(t)\psi(\mathbf r,t).$$

For the standard scalar Hamiltonian,

$$i\hbar \frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2\psi + V(\mathbf r,t)\psi.$$

The initial wavefunction and Hamiltonian domain determine an initial-value problem:

$$\psi(\mathbf r,t_0) = \psi_0(\mathbf r).$$

For a time-independent self-adjoint Hamiltonian with a discrete orthonormal eigenbasis,

$$\hat H\varphi_n = E_n\varphi_n,$$

an initial state expands as

$$\psi(\mathbf r,0) = \sum_n c_n\varphi_n(\mathbf r),$$

and evolves as

$$\psi(\mathbf r,t) = \sum_n c_n e^{-iE_nt/\hbar} \varphi_n(\mathbf r).$$

Each energy component acquires a phase. Relative phases change in time and can alter interference, even though the coefficient magnitudes $\lvert c_n\rvert$ remain fixed.

The Schrödinger Equation page owns the abstract dynamics, while Time-Dependent Schrödinger Equation develops coordinate-space solution methods.

Probability Conservation

For a real scalar potential, define

$$\varrho(\mathbf r,t) = \lvert\psi(\mathbf r,t)\rvert^2$$

and

$$\mathbf j(\mathbf r,t) = \frac{\hbar}{m} \operatorname{Im} \left( \psi^*\nabla\psi \right).$$

Combining the Schrödinger equation with its complex conjugate yields the local continuity equation

$$\frac{\partial\varrho}{\partial t} + \nabla\cdot\mathbf j =0.$$

Integrating over a fixed region $R$ and applying the divergence theorem gives

$$\frac{d}{dt} \int_R\varrho\,d^dr = -\int_{\partial R} \mathbf j\cdot d\mathbf S.$$

Probability in a region changes only through flux across its boundary. For the whole configuration space, boundary behavior that makes the net outward flux vanish gives

$$\frac{d}{dt} \int_\Omega \lvert\psi\rvert^2d\mu =0.$$

This is the position-space expression of norm preservation under unitary evolution. A complex absorbing potential or an explicitly non-Hermitian effective Hamiltonian can add source or sink terms; such dynamics is not closed-system unitary evolution on the stated Hilbert space.

The full derivations and interpretation are in Continuity Equation and Probability Current.

Boundary Conditions Complete the Hamiltonian

Consider the one-dimensional kinetic-energy expression

$$\hat T = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2}.$$

Integration by parts on an interval $[a,b]$ gives the boundary form

$$\begin{aligned} &\langle\phi\mid\hat T\psi\rangle - \langle\hat T\phi\mid\psi\rangle\\ &\qquad= -\frac{\hbar^2}{2m} \Big[ \phi^*\psi' - (\phi')^*\psi \Big]_a^b. \end{aligned}$$

A self-adjoint realization requires a domain whose boundary data make the appropriate boundary form vanish and whose adjoint has the same domain. Different admissible conditions describe different physical systems.

Common examples include:

• decay or scattering asymptotics on the full line;

• Dirichlet conditions $\psi(0)=\psi(L)=0$ for an ideal hard-wall box;

• periodic conditions

  $$\psi(L)=\psi(0), \qquad \psi'(L)=\psi'(0)$$

  for a ring without a boundary twist;

• Robin conditions, which relate a value and normal derivative at a boundary;

• continuity of $\psi$ and $\psi'$ across finite nonsingular potential steps;

• a derivative jump fixed by the coupling at a delta-function potential.

Boundary conditions are not decorative constraints added after solving the differential equation. They help define the operator, determine its spectrum, and enforce the intended probability flux.

The canonical practical treatment is Boundary Conditions.

Composition in Configuration Space

For two distinguishable particles,

$$\begin{aligned} \mathcal H_{AB} &= L^2(\Omega_A) \otimes L^2(\Omega_B)\\ &\cong L^2(\Omega_A\times\Omega_B). \end{aligned}$$

The joint wavefunction is

$$\Psi(\mathbf r_A,\mathbf r_B) = \langle \mathbf r_A,\mathbf r_B \mid\Psi\rangle.$$

A product state factorizes:

$$\Psi(\mathbf r_A,\mathbf r_B) = \psi_A(\mathbf r_A) \phi_B(\mathbf r_B).$$

Most joint wavefunctions do not factorize. A nonfactorizable pure-state wavefunction is entangled relative to the declared subsystem split.

Writing $d\mu_{AB}=d\mu_A\,d\mu_B$, the probability that $A$ lies in $R_A$ and $B$ lies in $R_B$ is

$$\Pr(R_A,R_B) = \int_{R_A\times R_B} \lvert\Psi\rvert^2 d\mu_{AB}.$$

For identical particles, physical wavefunctions are symmetric or antisymmetric under exchange:

$$\Psi(\mathbf r_1,\mathbf r_2) = \pm \Psi(\mathbf r_2,\mathbf r_1),$$

with the plus sign for bosons and the minus sign for fermions when the displayed variables include all relevant one-particle degrees of freedom.

Spin adds discrete indices. A spin-$1/2$ position-space state, for example, is a spinor

$$\psi(\mathbf r) = \begin{pmatrix} \psi_\uparrow(\mathbf r)\\ \psi_\downarrow(\mathbf r) \end{pmatrix},$$

normalized by summing component densities:

$$\int d^3r \left( \lvert\psi_\uparrow\rvert^2 + \lvert\psi_\downarrow\rvert^2 \right) =1.$$

These are coordinate forms of tensor products, not exceptions to the abstract composition postulate.

Crosswalk to the Abstract Postulates

The wave-mechanics statements translate the abstract formalism as follows:

• A ray $\lvert\psi\rangle$ becomes a wavefunction $\psi(\mathbf r)=\langle\mathbf r\mid\psi\rangle$.
• The abstract inner product becomes an integral against the coordinate measure.
• A self-adjoint observable becomes a multiplication, differential, or integral operator with a specified domain.
• A position-region projector becomes multiplication by the indicator of that region, expressed formally through generalized position kets.
• The Born trace or expectation rule becomes an integral of a probability density.
• Unitary evolution becomes the time-dependent Schrödinger partial differential equation.
• Tensor products become functions on product configuration spaces, possibly with spinor indices or exchange symmetry.

Changing representation does not change predictions. For example,

$$\langle\phi\mid\psi\rangle = \int d^dr\, \phi(\mathbf r)^* \psi(\mathbf r)$$

and, with a unitary Fourier convention,

$$\langle\phi\mid\psi\rangle = \int d^dp\, \widetilde\phi(\mathbf p)^* \widetilde\psi(\mathbf p).$$

The coordinate and momentum wavefunctions are different descriptions of the same Hilbert-space vectors.

Worked Example: The Infinite Square Well as a Package

Consider one particle confined to $0<x<L$ by ideal hard walls. The Hilbert space is

$$\mathcal H = L^2(0,L),$$

and the Hamiltonian acts as

$$\hat H = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2}$$

on a domain with Dirichlet conditions

$$\psi(0)=\psi(L)=0.$$

The normalized stationary eigenfunctions and energies are

$$\varphi_n(x) = \sqrt{\frac{2}{L}} \sin\frac{n\pi x}{L},$$

and

$$E_n = \frac{n^2\pi^2\hbar^2} {2mL^2}, \qquad n=1,2,\ldots.$$

An arbitrary normalized state in the discrete span has

$$\psi(x,0) = \sum_{n=1}^{\infty} c_n\varphi_n(x), \qquad \sum_{n=1}^{\infty} \lvert c_n\rvert^2 =1,$$

and evolves as

$$\psi(x,t) = \sum_{n=1}^{\infty} c_n e^{-iE_nt/\hbar} \varphi_n(x).$$

For the ground state,

$$\psi_1(x,t) = \sqrt{\frac{2}{L}} \sin\frac{\pi x}{L} e^{-iE_1t/\hbar}.$$

Its position density is time independent because the phase has unit magnitude:

$$\lvert\psi_1(x,t)\rvert^2 = \frac{2}{L} \sin^2\frac{\pi x}{L}.$$

The Born probability for the left half is

$$\begin{aligned} \Pr(0<x<L/2) &= \frac{2}{L} \int_0^{L/2} \sin^2\frac{\pi x}{L}\,dx\\ &= \frac12. \end{aligned}$$

The current vanishes because the spatial eigenfunction can be chosen real:

$$j(x,t) = \frac{\hbar}{m} \operatorname{Im} \left( \psi_1^* \frac{\partial\psi_1}{\partial x} \right) =0.$$

This one model uses the Hilbert space, operator domain, boundary conditions, normalization, spectrum, unitary time dependence, position Born rule, and current check. The ideal walls are a model assumption; a finite physical barrier requires a different Hamiltonian and matching problem.

A Wave-Mechanics Workflow

For a new problem:

1. Declare configuration space. Is the particle on a line, interval, ring, plane, or three-dimensional region?
2. Declare the measure and components. Include coordinate Jacobians and spinor indices.
3. Specify the Hamiltonian as an operator. State its differential expression, domain, and boundary or asymptotic conditions.
4. Choose and normalize the initial state.
5. Solve the spectral or initial-value problem.
6. Represent the measurement. For position, specify a detector region or response; for another observable, use its spectral data or effects.
7. Compute probabilities and expectations.
8. Check normalization, dimensions, boundary flux, and limiting cases.

The Hamiltonian is physical model input. The postulates explain how a stated Hamiltonian generates predictions; they do not derive its potential or boundary conditions from notation alone.

Consistency Checks

Normalization

Verify

$$\int_\Omega \lvert\psi(\mathbf r,t)\rvert^2 d\mu =1$$

at the initial time and after evolution. For exact self-adjoint evolution, later normalization follows theoretically; in a numerical calculation, drift diagnoses discretization or time-stepping error.

Units

Check that $\lvert\psi\rvert^2d\mu$ and every reported probability are dimensionless. Fourier conventions give position and momentum wavefunctions different physical units.

Boundary form and flux

Confirm that the stated boundary conditions support the claimed self-adjoint operator and that the probability flux matches the physical boundary. A hard wall should not leak.

Limiting behavior

Check free-particle, weak-potential, large-box, short-time, or symmetry limits as appropriate. A formally normalized answer can still solve the wrong boundary-value problem.

Representation agreement

When practical, compute a quantity in more than one representation. Position-space integration and expansion in an orthonormal eigenbasis should agree.

What a Scalar Wavefunction Does Not Cover

A single scalar $\psi(\mathbf r,t)$ is insufficient for:

• spin and other internal components;
• mixed preparations and reduced subsystem states;
• several particles without a configuration-space wavefunction;
• identical-particle exchange symmetry unless it is imposed;
• variable particle number;
• relativistic quantum field theory;
• general measurements whose detector response is not a sharp position region.

These cases extend the representation; they do not invalidate the abstract postulates. The danger is not using wave mechanics, but mistaking its simplest scalar form for the whole theory.

Common Mistakes

1. Treating $\psi$ as a classical material wave. It is a complex probability-amplitude representation.
2. Calling $\lvert\psi(\mathbf r)\rvert^2$ a probability. It is a density; integrate it over a region.
3. Forgetting the coordinate measure. Jacobian factors are part of normalization and expectation values.
4. Assigning physical meaning to one pointwise value of an $L^2$ representative. States are defined almost everywhere.
5. Using generalized eigenfunctions as normalized states. Position kets and plane waves require distributional normalization or wave packets.
6. Writing a differential expression without a domain. Boundary conditions can change the operator and spectrum.
7. Confusing symmetric with self-adjoint. Domain equality matters for unbounded operators.
8. Dropping relative phases. Global phase is irrelevant; relative phase controls interference and current.
9. Assuming stationary means time-independent wavefunction. A stationary energy eigenstate carries a time-dependent phase while its density is time independent.
10. Using one scalar wavefunction for every system. Spin, mixed states, many particles, and fields require richer structures.

Connections

• Wavefunctions as Representations explains why a wavefunction depends on the chosen generalized basis.
• Coordinate Representation develops operator kernels and coordinate-space notation.
• Wavefunctions and Probability Density owns the practical interpretation of spatial densities.
• Finite-Dimensional Postulates gives the complementary matrix-first formulation.
• Equivalent Formulations explains why representation changes preserve empirical predictions.
• Boundary Conditions develops self-adjoint boundary-value problems.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018 — standard wave-mechanics development and examples.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994 — Hilbert-space foundations connected carefully to coordinate representations.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020 — abstract and wave-mechanics formulations with symmetries and spin.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vol. 1, Wiley, 1977 — detailed wavefunction, operator, and measurement treatment.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013 — rigorous bridge between Hilbert-space quantum theory and differential operators.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, revised and enlarged ed., Academic Press, 1980 — domains, self-adjointness, and spectral foundations.

Exercises

1. Normalize a Gaussian. In one dimension, let

$$\psi(x) = A \exp\!\left( -\frac{x^2}{4\sigma^2} \right), \qquad \sigma>0.$$

Find $\lvert A\rvert$ and state its physical units.

Solution

The density is

$$\lvert\psi(x)\rvert^2 = \lvert A\rvert^2 \exp\!\left( -\frac{x^2}{2\sigma^2} \right).$$

Using the Gaussian integral,

$$\int_{-\infty}^{\infty} e^{-x^2/(2\sigma^2)}dx = \sqrt{2\pi}\,\sigma.$$

Normalization gives

$$\lvert A\rvert^2 \sqrt{2\pi}\,\sigma =1,$$

so

$$\lvert A\rvert = \frac{1} {(2\pi\sigma^2)^{1/4}}.$$

In one dimension, $A$ and $\psi$ have units $\mathsf L^{-1/2}$. The phase of $A$ is a global phase and may be chosen conveniently.

2. Use the spherical measure. A normalized spherically symmetric state has wavefunction $\psi(\mathbf r)=Ce^{-r/a}$ with $a>0$. Find $\lvert C\rvert$.

Solution

Spherical symmetry gives

$$1 = 4\pi\lvert C\rvert^2 \int_0^\infty r^2e^{-2r/a}\,dr.$$

Using

$$\int_0^\infty r^2e^{-\beta r}\,dr = \frac{2}{\beta^3}$$

with $\beta=2/a$ gives

$$\int_0^\infty r^2e^{-2r/a}\,dr = \frac{a^3}{4}.$$

Therefore,

$$\pi a^3\lvert C\rvert^2 =1,$$

and

$$\lvert C\rvert = \frac{1}{\sqrt{\pi a^3}}.$$

Omitting the factor $4\pi r^2$ would give the wrong normalization and wrong units.

3. Global and relative phase. Compare

$$\psi_\alpha = e^{i\alpha} \frac{\psi_1+\psi_2}{\sqrt2}$$

with

$$\psi_\beta = \frac{\psi_1+e^{i\beta}\psi_2}{\sqrt2},$$

where $\psi_1$ and $\psi_2$ overlap. Which phase can affect the position density?

Solution

The common factor in $\psi_\alpha$ cancels:

$$\lvert\psi_\alpha\rvert^2 = \frac12 \lvert\psi_1+\psi_2\rvert^2.$$

It is a global phase. For $\psi_\beta$,

$$\begin{aligned} \lvert\psi_\beta\rvert^2 &= \frac12 \Big( \lvert\psi_1\rvert^2 + \lvert\psi_2\rvert^2 \Big)\\ &\quad+ \operatorname{Re} \left( e^{-i\beta}\psi_1\psi_2^* \right). \end{aligned}$$

The relative phase $\beta$ changes the interference term wherever the components overlap.

4. A boundary-form check. On $[0,L]$, show that the kinetic-energy boundary form vanishes for two twice-differentiable functions satisfying Dirichlet conditions at both endpoints.

Solution

The boundary form is proportional to

$$\Big[ \phi^*\psi' - (\phi')^*\psi \Big]_0^L.$$

Dirichlet conditions give

$$\begin{aligned} \phi(0)&=\phi(L)=0,\\ \psi(0)&=\psi(L)=0. \end{aligned}$$

At each endpoint, the first term vanishes because $\phi=0$ and the second because $\psi=0$. Hence the entire boundary form vanishes. This establishes symmetry on the stated domain; a full self-adjointness claim also requires comparing the adjoint domain.

5. Periodic quantization. A free particle on a ring of circumference $L$ has $\psi(x+L)=\psi(x)$. Starting from $\psi_k(x)=Ae^{ikx}$, find the allowed $k$, normalize the wavefunction, and give the energies.

Solution

Periodicity requires

$$e^{ikL}=1,$$

so

$$k_n = \frac{2\pi n}{L}, \qquad n\in\mathbb Z.$$

Normalization on one period gives

$$1 = \lvert A\rvert^2L,$$

so one may choose $A=1/\sqrt L$. The energies are

$$E_n = \frac{\hbar^2k_n^2}{2m} = \frac{2\pi^2\hbar^2n^2} {mL^2}.$$

Unlike a plane wave on the full line, the periodic wavefunction is normalizable on its finite configuration space.

6. Continuity equation in one dimension. For real $V(x,t)$, start from the Schrödinger equation and its complex conjugate and show

$$\frac{\partial}{\partial t} \lvert\psi\rvert^2 + \frac{\partial j}{\partial x} =0,$$

where

$$j = \frac{\hbar}{m} \operatorname{Im} \left( \psi^* \frac{\partial\psi}{\partial x} \right).$$

Solution

Multiply the Schrödinger equation by $\psi^*$ and its conjugate by $\psi$, then subtract. The real potential terms cancel:

$$\begin{aligned} i\hbar \frac{\partial}{\partial t} \lvert\psi\rvert^2 &= -\frac{\hbar^2}{2m} \left( \psi^*\psi'' - \psi(\psi^*)'' \right). \end{aligned}$$

The spatial expression is a total derivative:

$$\psi^*\psi'' - \psi(\psi^*)'' = \frac{\partial}{\partial x} \left( \psi^*\psi' - \psi(\psi^*)' \right).$$

Dividing by $i\hbar$ and using

$$j = \frac{\hbar}{2mi} \left( \psi^*\psi' - \psi(\psi^*)' \right)$$

gives the continuity equation.

7. A two-state box superposition. In the infinite square well, let

$$\psi(x,0) = \frac{ \varphi_1(x)+\varphi_2(x) }{\sqrt2}.$$

Write $\psi(x,t)$ and identify the angular frequency of the interference term in $\lvert\psi(x,t)\rvert^2$.

Solution

Each eigenstate acquires its own phase:

$$\begin{aligned} \psi(x,t) &= \frac1{\sqrt2} \Big[ e^{-iE_1t/\hbar}\varphi_1(x)\\ &\qquad+ e^{-iE_2t/\hbar}\varphi_2(x) \Big]. \end{aligned}$$

Because the box eigenfunctions can be chosen real,

$$\begin{aligned} \lvert\psi(x,t)\rvert^2 &= \frac12 \left[ \varphi_1(x)^2+\varphi_2(x)^2 \right]\\ &\quad+ \varphi_1(x)\varphi_2(x)\\ &\qquad\times \cos\!\left( \frac{E_2-E_1}{\hbar}t \right). \end{aligned}$$

The interference oscillates with angular frequency

$$\omega_{21} = \frac{E_2-E_1}{\hbar} = \frac{3\pi^2\hbar} {2mL^2}.$$

The total norm remains one even though the spatial density changes.

8. Factorization and entanglement. Let $\phi_0$ and $\phi_1$ be orthonormal one-particle wavefunctions. Show that

$$\begin{aligned} \Psi(x_A,x_B) &= \frac1{\sqrt2} \Big[ \phi_0(x_A)\phi_0(x_B)\\ &\qquad+ \phi_1(x_A)\phi_1(x_B) \Big]. \end{aligned}$$

cannot be written as $f(x_A)g(x_B)$.

Solution

Suppose a product form existed. Expand

$$\begin{aligned} f&=a\phi_0+b\phi_1+\cdots,\\ g&=c\phi_0+d\phi_1+\cdots. \end{aligned}$$

Matching the four coefficients in the displayed two-dimensional subspace would require

$$ac=\frac1{\sqrt2}, \qquad bd=\frac1{\sqrt2},$$

and

$$ad=0, \qquad bc=0.$$

The first two equations require $a,b,c,d$ all to be nonzero, contradicting the last two. Hence the wavefunction is nonfactorizable and the two particles are entangled relative to the $A$–$B$ split.