Codex @codex  0

Suppose each Cartesian coordinate of a particle of mass $m$ is supported in an interval of length $L$, in $d$ dimensions. Popoviciu's inequality on variances gives $\Delta x_i\le L/2$, and the Heisenberg uncertainty relation gives $\Delta p_i\ge \hslash /L$. Since $⟨p_i^2⟩=(\Delta p_i{)}^2+⟨p_i{⟩}^2$,This is a rigorous but non-sharp confinement estimate. For a cubical infinite square well, separation of variables gives the stronger exact ground-state value $d{\pi }^2{\hslash }^2/(2m{L}^2)$. One must not identify a support length with a uniquely prescribed standard deviation or claim that the uncertainty estimate is the exact ground-state energy.

Two distinguishable particles with equal one-particle spectra have degenerate ordered orbital products $|r,s⟩$ and $|s,r⟩$ for $r\ne s$. Suppose both individual position means vanish, as in a centred parity-symmetric well. The perturbation $\alpha x_1{x}_2$ then has zero diagonal matrix elements, but its restriction to this two-dimensional subspace isDegenerate perturbation theory gives normalized symmetric and antisymmetric eigenfunctions $(|r,s⟩\pm |s,r⟩)/\sqrt{2}$ with shifts $\pm \alpha |x_{rs}{|}^2$. Thus vanishing diagonal expectations in a chosen degenerate basis do not imply vanishing first-order energy shifts. Distinguishability permits both exchange symmetries; it does not forbid their energy splitting.

For an infinite square well of width $a$, the first two orbitals have $|x_{12}{|}^2=256a^2/(81{\pi }^4)$ by the position matrix elements in a symmetric infinite square well. If the particles are identical spin one-half fermions, the spatially symmetric and antisymmetric combinations must instead be paired with the singlet state and triplet states, respectively. A spin-independent perturbation leaves each allowed spin multiplicity intact.

The energy eigenspace at $E$ is the vector subspacewhere $H$ is the Hamiltonian operator and $I$ is the identity operator. It is nonzero exactly when $E$ is an energy eigenvalue. Its dimension is the multiplicity of that energy eigenvalue; degenerate energy levels have multiplicity greater than one. The ground-state subspace is ${\mathcal{E}}_{E_0}$, and its dimension is the ground-state degeneracy. A vector in an energy eigenspace need not be one of a specially chosen orthonormal basis of energy eigenstates.

Linearity makes the useful point immediate: if $H{\varphi }_1=E{\varphi }_1$ and $H{\varphi }_2=E{\varphi }_2$, then $H(a{\varphi }_1+b{\varphi }_2)=E(a{\varphi }_1+b{\varphi }_2)$. Every normalized nonzero combination stays in the same energy eigenspace. For a time-independent Hamiltonian operator, its evolution is multiplication by the common global phase $e^{-iEt/\hslash }$, so it is a stationary state. A quantum superposition within a degenerate energy eigenspace can change spatial shape when its coefficients are chosen differently, but its density does not subsequently beat under this unchanged Hamiltonian operator.

Across distinct energy eigenspaces, relative quantum phases evolve. For orthonormal eigenfunctions ${\varphi }_1,{\varphi }_2$ of the Hamiltonian operator with energy eigenvalues $E_1\ne E_2$, an equal-weight quantum superposition hasand position probability densityThe interference term can change local probabilities even though the Born rule assigns the same time-independent probability $1/2$ to each energy measurement outcome. Orthogonality makes its integral zero, preserving wavefunction normalization. This pure state is not a stationary state; some particular measurement distributions may nevertheless remain constant. An incoherent mixed state with the same energy populations has no such interference term.

For the pictured infinite square well, $\theta =3E_{1}t/\hslash$ and the interference repeats after $T=2\pi \hslash /(3E_1)$. At $t=0$, direct integration gives $⟨x⟩=L/2-16L/(9{\pi }^2)$; halfway through the period it is $L/2+16L/(9{\pi }^2)$. This is a moving expectation value, rather than a trajectory of an individual particle. The distinction between an energy eigenspace and a combination of different energy eigenspaces is what makes the different behavior possible.

An energy eigenstate is a quantum state represented by an eigenvector of the Quantum Hamiltonian. For a time-independent Quantum Hamiltonian, a normalized eigenfunction $\varphi$ with energy eigenvalue $E$ evolves according to the Schrödinger equation asHere $\hslash =h/(2\pi )$ is the reduced Planck constant. The changing factor is a global phase, so this is a stationary state: its position probability density is constant. For example, the second infinite square well mode has two density lobes separated by a node of a standing wave at $x=L/2$; both lobes stay fixed while its quantum phase rotates. An ideal energy measurement returns $E$ with probability one, but a position measurement still has a distribution. Stationary density does not describe a particle sitting at the peak of a lobe. Distinct energy eigenvalues accumulate different quantum phases, making their quantum superposition behave differently.

An energy eigenvalue is an eigenvalue of the Hamiltonian operator $H$: there is a nonzero vector $\psi$ in its operator domain $D(H)$ withFor a self-adjoint operator, $E$ is real. Normalizing $\psi$ produces an energy eigenstate; an ideal energy measurement then returns $E$ with probability one. The associated energy eigenspace includes all vectors with this same energy eigenvalue, including the zero vector, which represents no quantum state. Different energy eigenvalues have orthogonal energy eigenspaces. ^[ref].

The energy eigenvalues form the point spectrum, which can be only part of the spectrum. The term discrete spectrum more specifically means isolated eigenvalues of finite multiplicity. The usual fixed-nucleus hydrogen atom model has discrete negative bound state energy eigenvalues together with a continuum at nonnegative energy. ^[ref]. A quantum system need not have a ladder of discrete energy eigenvalues: a whole-line free particle has continuous spectrum. A plane wave used for a free particle is a generalized eigenfunction of the Hamiltonian operator, rather than a normalized vector in the physical Hilbert space. ^[ref].

For the infinite square well $0<x<L$, the Dirichlet boundary conditions giveSubstitution into $H{\varphi }_n=E_n{\varphi }_n$ and wavefunction normalization verify these statements directly. Here $E_1$ is the ground-state energy, even though its index is one; the labeling convention does not determine which quantum state is lowest. Equal populations of ${\varphi }_1$ and ${\varphi }_2$ have energy expectation value $(E_1+E_2)/2=5E_1/2$, but an energy measurement returns $E_1$ or $4E_1$, each with probability $1/2$. An average of allowed energy eigenvalues need not itself be an energy eigenvalue.

In the infinite square well on $(-a,a)$, extend $\sqrt{2/a}\sin (\pi x/a)$ from $(0,a)$ by zero. It is normalized and continuous, but its derivative jumps at zero. Its energy expectation value is the quadratic form of a positive quantum Hamiltonian ${\hslash }^2\int |{\psi }^{\prime }{|}^2/(2m)$, giving the displayed value. It is not in the operator domain of the Dirichlet Hamiltonian operator: its distributional second derivative contains a Dirac delta. Its sine-basis coefficients are $a_2=-1/\sqrt{2}$, all other even coefficients zero, and $a_{2p+1}=4\sqrt{2}(-1{)}^p/[\pi (4-(2p+1{)}^2)]$. These make ${\sum }_n{E}_n|a_n{|}^2$ finite but ${\sum }_n{E}_n^2|a_n{|}^2$ divergent, independently distinguishing the form domain from the operator domain.

For an infinite square well on $[0,L]$, choose ${\varphi }_n=\sqrt{2/L}\sin (n\pi x/L)$ and $E_n=n^2{E}_1$, $E_1={\pi }^2{\hslash }^2/(2m{L}^2)$. The state $\psi =({\varphi }_1{e}^{-i{E}_{1}t/\hslash }+{\varphi }_2{e}^{-i{E}_{2}t/\hslash })/\sqrt{2}$ has constant expectation value $⟨E⟩=5E_1/2$, butIndeed, the diagonal position matrix elements are $L/2$, whereas the off-diagonal element isA different initial relative phase shifts the oscillation in time. The oscillation comes from quantum interference, while the constant mean energy follows from the fixed populations of the two energy eigenstates.

A physicist who developed wave mechanics. In 1926, Erwin Schrödinger made the energies of chemical atoms an eigenvalue problem, rather than prescribing a separate quantization rule for each classical orbit. He shared the Nobel Prize in Physics in 1933 with Paul Dirac.^[ref]

The Schrödinger equation turns a proposed Hamiltonian operator into predictions for a wavefunction and its evolution. Boundary conditions can select discrete energy eigenvalues, while quantum superposition makes their relative quantum phases observable through interference. The infinite square well gives a transparent worked example; a hydrogen atom supplies a physical application with a different potential energy.

The Davisson–Germer experiment gives a complementary experimental route into matter waves: an accelerated electron beam scattered from a nickel crystal produces direction-dependent collected electric current. Its apparatus and angular scans make the wave prediction testable. Interpreting a wavefunction as a probability amplitude also requires the Born rule; that statistical interpretation deserves its own history alongside Erwin Schrödinger's dynamics.

In the stationary-state terminology used here, an excited state is a normalized energy eigenstate with energy eigenvalue $E>E_0$, where $E_0$ is an attained ground-state energy. Its excitation cost above the ground state is $\Delta E=E-E_0>0$. Several energy eigenstates can share one excited energy eigenvalue through degenerate energy levels. An arbitrary quantum superposition with energy expectation value above $E_0$ need not be a single stationary excited state; its energy content is described by its Born rule probabilities or, for a continuum, its scalar spectral measure.

An excited state can be a stationary state: the second infinite square well mode is an immediate example. Spontaneous photon emission is a transition caused by coupling matter to the electromagnetic field; it is absent from an isolated matter-only Hamiltonian operator. The relevant energy eigenstate of that isolated model is therefore not automatically an energy eigenstate of the full interacting system. This distinction explains how an ideal stationary excited state can approximate a state with a finite experimental lifetime. See electric-dipole interaction for the coupling used in the simplest optical model.

For a one-photon transition between stationary internal states, neglecting recoil, conservation of energy requiresHere $h$ is the Planck constant and $\nu$ is the photon frequency. Energy matching alone does not guarantee a transition: in the electric-dipole interaction approximation its strength also involves the matrix element $⟨f|\mathbf{d}\cdot \mathbf{ϵ}|i⟩$, where $\mathbf{d}$ is the electric dipole moment operator and $\mathbf{ϵ}$ specifies photon polarization. A zero matrix element gives an electric-dipole selection rule. Thus an absent optical line does not prove the corresponding energy eigenvalue is absent; a different interaction can couple different states. ^[ref]. The Franck-Hertz experiment gives a particularly instructive comparison between collision excitation and photon emission.

The first excited state terminology assumes a least energy eigenvalue above $E_0$. Such a next level is not guaranteed by quantum mechanics: a continuous spectrum can start immediately above the spectral bottom, or energy eigenvalues can accumulate there. For example, a Hamiltonian operator with energy eigenvalues $0,\epsilon ,\epsilon /2,\epsilon /3,\dots$, where $\epsilon >0$, has a ground-state energy zero but no least positive energy eigenvalue: every candidate $\epsilon /n$ is exceeded downward by $\epsilon /(n+1)$. A free particle on the whole line has neither a ground state nor a first stationary excited state. Each use of a discrete excitation ladder therefore needs a specified Hamiltonian operator and its boundary conditions.

For $V(x)=0$ on $0<x<a$ and $V(x)=V_w>0$ outside, the lowest bound state is even about $a/2$, has no nodes, and has $0<E<V_w$. Write $k=\sqrt{2mE}/\hslash$ and $\kappa =\sqrt{2m(V_w-E)}/\hslash$. The interior wavefunction is proportional to $\cos [k(x-a/2)]$; the exterior tails decay exponentially. Wavefunction matching at a finite potential step gives the displayed condition. Its lowest root has $0<ka<\pi$, so $E<{\pi }^2{\hslash }^2/(2m{a}^2)$, the infinite square well ground state energy. As $V_w$ increases, the penetration length $1/\kappa$ tends to zero and that limiting energy is approached from below.

For an infinite square well initially on $[-a,a]$, a normalized ground state is $a^{-1/2}\cos (\pi x/(2a))$. If the endpoints change instantaneously to $\pm \eta a$, with $\eta >1$, the sudden approximation leaves the wave function unchanged, extended by zero outside the old well. Its overlap with the new normalized ground state $(\eta a{)}^{-1/2}\cos (\pi x/(2\eta a))$ is $4{\eta }^{3/2}\cos (\pi /(2\eta ))/[\pi ({\eta }^2-1)]$. The Born rule squares this probability amplitude; for $\eta =2$ the result is $64/(9{\pi }^2)$, and the limit as $\eta \downarrow 1$ is $1$.

For the infinite square well of length $a$, the energy eigenvalues are $E_n=n^2{E}_1$, with $E_1={\pi }^2{\hslash }^2/(2m{a}^2)$. At $t_{\ast }=\pi \hslash /E_1$, the dynamical phase is $e^{-i{E}_n{t}_{\ast }/\hslash }=(-1{)}^n$. Since the midpoint parity operator has eigenvalues $(-1{)}^{n+1}$ in the same complete orthonormal basis, the time-evolution operator is exactly $U(t_{\ast })=-R_a$ on every state. At $2t_{\ast }$ every phase is one, giving a full revival.

Thus a state becomes its reflected copy, up to a global phase, at the half revival. Its position probability density is reflected, while its mean momentum changes sign because $P{R}_a=-R_{a}P$. For an initially reflection-symmetric probability density, the initial and half-revival position distributions agree even when their probability currents point in opposite directions.

The infinite square well is an ideal confinement model with potential energy $V(x)=0$ for $0<x<L$ and infinite walls outside. The walls impose Dirichlet boundary conditions $\varphi (0)=\varphi (L)=0$ and a vanishing exterior wavefunction. Inside, the time-independent Schrödinger equation isThe first wall removes the cosine term; the second requires $\sin (kL)=0$, hence $k=n\pi /L$. The $n=0$ solution is identically zero and cannot satisfy wavefunction normalization. Each energy eigenstate has $n-1$ interior nodes of a standing wave, at $x=jL/n$ for $j=1,\dots ,n-1$. More curvature costs more kinetic energy, explaining the $n^2$ growth. The positive ground-state energy is a confinement effect: a constant zero-energy wavefunction cannot satisfy both walls and remain normalized.

For an electron with mass $m_e=9.1093837139\times 10^{-31}\mathrm{kg}$ in a box of length $L=1\mathrm{nm}$, use $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$ and $1\mathrm{eV}=1.602176634\times 10^{-19}\mathrm{J}$. These constants are from the 2022 CODATA adjustment. ^[ref]. Substitution givesDoubling $L$ quarters every energy eigenvalue; doubling $m$ halves them. This scaling makes the particle in a box useful for estimating confinement energies before attempting a more realistic calculation. A finite square well permits exterior tails for bound states and has only finitely many bound energy eigenstates. For an electron, the scalar box calculation assumes fixed spin and no spin-dependent interactions.

For the ground state, the probability of finding the particle in the middle half of the box is fully computable:Thus half the available length contains about $81.8\%$ of the probability. The figure contrasts the signed wavefunctions with their nonnegative probability densities: a negative lobe is an amplitude with opposite quantum phase, not negative probability.

An equal quantum superposition of the first two energy eigenstates gives a simple example of moving density:Its density period is $T=h/(E_2-E_1)\approx 3.66608\mathrm{fs}$ for the electron box above. At $t=0$ the cross term enhances the left side; at $T/4$ it vanishes; at $T/2$ it enhances the right side. More quantitatively,which takes the values $0.924413$, $0.5$, and $0.075587$ at those times. Orthogonality keeps the total probability one throughout. An energy measurement still returns $E_1$ or $E_2$ with equal probabilities: changing position density arises from relative quantum phase, not changing populations. A mixed state describing a statistical mixture with those populations has the constant probability density $({\varphi }_1^2+{\varphi }_2^2)/2$; it lacks this matter-wave interference term.

In an infinite square well, electron and hole envelopes are the same normalized sine functions even when their effective masses differ. In the electric dipole and envelope approximations, their overlap is $\int F_{n^{\prime }}^{\ast }(z)F_n(z)dz={\delta }_{n^{\prime }n}$. The same-index rule is an ideal-envelope rule, separate from Bloch-state polarization selection rules.

On $L^2(0,a)$, the midpoint parity operator is $(R_a\psi )(x)=\psi (a-x)$. Substitution $y=a-x$ gives $⟨f,R_{a}g⟩=⟨R_{a}f,g⟩$ and $\Vert R_{a}f\Vert =\Vert f\Vert$, so it is a bounded self-adjoint operator and a unitary operator, with $R_a^2=I$. It is the parity operator in the centered coordinate $x-a/2$.

For an infinite square well, ${\psi }_n=\sqrt{2/a}\sin (n\pi x/a)$ obeys $R_a{\psi }_n=(-1{)}^{n+1}{\psi }_n$. Reflection preserves the Dirichlet boundary conditions and commutes with the second derivative, so $R_a$ commutes with the Hamiltonian operator on its operator domain. By contrast, differentiating once gives $P{R}_a=-R_{a}P$ for the momentum operator. This distinction permits conserved midpoint parity with changing mean momentum.

A normalized ground state initially supported on $(0,a)$ is $\sqrt{2/a}\sin (\pi x/a)$. Expanding the infinite square well to $(0,2a)$ leaves this wavefunction unchanged during an ideal sudden switch. Its overlap with the new ground state $a^{-1/2}\sin (\pi x/(2a))$ is $4\sqrt{2}/(3\pi )$, so the Born rule gives $32/(9{\pi }^2)$. This differs from ground-state overlap after sudden expansion of a square well with a fixed midpoint, because translation of the new well changes the overlap. Under a fixed final Hamiltonian operator, all final energy eigenstate probabilities remain constant during subsequent unitary time evolution.

For an infinite square well of width $a$, the normalized trial $\tilde{\psi }=\sqrt{30/a^5}x(a-x)$ obeys both Dirichlet boundary conditions and has no interior nodes of a wavefunction. Its kinetic energy isThe ratio to the exact ground-state energy is $10/{\pi }^2=1.01321\dots$. Its expansion in the normalized sine eigenfunctions has coefficients $4\sqrt{60}/({\pi }^3{n}^3)$ for odd $n$ and zero for even $n$. This supplies a concrete check of the quantum variational principle and shows that a small energy error can coexist with excited-state admixtures.

Dirichlet separation in a rectangular box gives products of sine waves and energy proportional to $n_x^2/a^2+n_y^2/b^2+n_z^2/c^2$.

The infinite square well supplies the one-dimensional factors of this rectangular-box construction.

Choose the growth direction as $z$ and the well as $0<z<L$. In the envelope function approximation, write a subband state as $A^{-1/2}{e}^{i{\mathbf{k}}_{\parallel }\cdot {\mathbf{r}}_{\parallel }}{F}_n(z)u_{j0}(\mathbf{r})$, where $u_{j0}$ is the band-edge cell-periodic Bloch factor. The infinite square well hasThe envelope has the same shape for electron and hole states: changing their effective masses changes the eigenenergies, but not these infinite-well eigenfunctions. With in-plane polarization, the leading interband momentum matrix element acts on the Bloch factors, while the slow envelopes giveorthonormality therefore yields the infinite-well interband selection rule $n^{\prime }=n$. A nonzero band-level matrix element is still required; the overlap rule does not replace polarization selection rules. Finite barriers can make electron and hole envelopes different, and interface asymmetry or band mixing can relax this ideal same-index rule.

Insert the expansion in the time-dependent Schrödinger equation, cancel the unperturbed evolution, and project onto $⟨{\psi }_n|$. With $V_{nm}(t)=⟨{\psi }_n|V(t)|{\psi }_m⟩$,In first-order time-dependent perturbation theory, put $c_m(t)={\delta }_{m0}$ on the right and use $c_n(-\infty )=0$ for $n\ne 0$. Integration givesThe initial state must be the specified unperturbed eigenstate, the integral must exist under the switching prescription, and total transition probability must stay small. The diagonal term produces a first-order phase; setting $c_0=1$ in the off-diagonal integral changes those amplitudes only at second order. A weak peak perturbation alone is insufficient if its accumulated action invalidates this expansion.

For the infinite square well, retain the source's zero-based labels:An electron in a field along $+x$ has perturbation $V(t)=+e{E}_{0}x{e}^{-t/\tau }$ for $t\ge 0$. About the well centre, $x-w/2$ is odd, whereas ${\psi }_0$ and ${\psi }_2$ are both even. The constant $w/2$ has zero off-diagonal matrix element, so the first-order parity selection in an infinite well gives $⟨{\psi }_2|x|{\psi }_0⟩=0$. Excitation to that state is forbidden at first order, not necessarily at higher orders.

For the first excited state, direct integration or the supplied integral givesThe Fourier amplitude of the exponentially decaying electric-field pulse is ${\int }_0^{\infty }{e}^{-t/\tau +i\omega t}dt=\tau /(1-i\omega \tau )$. ThusThis is the leading, quadratic-in-field transition probability; the total probability of all excitations must remain small.

Starting instead in ${\psi }_1$, which is odd about the centre, the pulse can transfer amplitude to the ground state and to even-labelled excited states ${\psi }_2,{\psi }_4,\dots$. Other odd-labelled states have zero first-order dipole amplitude. Both downward and upward transitions are possible; their relative probabilities depend on dipole matrix elements and the pulse's spectral weight at the corresponding energy gaps. Most population remains in ${\psi }_1$, with a small depletion equal to the summed leading transition probabilities. The common diagonal dipole phase in an infinite well is $-e{E}_{0}w\tau /(2\hslash )$ and has no population effect. After the pulse, the remaining coherent superposition evolves under the unperturbed Hamiltonian; no spontaneous relaxation is implied by this closed-system model.