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A stationary evanescent wave does not assign a classical trajectory or a real local speed inside the forbidden region. The dwell time in a semi-infinite potential step measures probability stored in $x>0$ divided by incident probability current. For incident amplitude one,This gives a time in terms of $m,E,V_0$ through the definitions of $k,\kappa$. A simpler dimensional estimate is the penetration length divided by the incident speed, $\Delta x/v_{\mathrm{in}}=m/(\hslash k\kappa )=\hslash /[2\sqrt{E(V_0-E)}]$. The dwell time equals $(2E/V_0)$ times that estimate. Different operational timing definitions need not agree; treating the imaginary under-barrier momentum as a real particle velocity would not justify a unique traversal time.

Assume a real, time-independent potential with the same zero asymptote on both sides, so $k=\sqrt{2mE}/\hslash$ is the same incident and transmitted wave number. For unit incoming amplitudes,Multiplying the two Schrödinger equations crosswise and subtracting shows that the Wronskian $W={\psi }_L{\psi }_R^{\prime }-{\psi }_L^{\prime }{\psi }_R$ is constant. Its two asymptotic values are $-2ik{t}_R$ and $-2ik{t}_L$, hence $t_L=t_R$.

For a real potential the probability current $j=(\hslash /m)\mathrm{Im}({\psi }^{\ast }{\psi }^{\prime })$ is conserved, giving $|r_L{|}^2+|t_L{|}^2=|r_R{|}^2+|t_R{|}^2=1$. ThereforeThe reflection amplitudes can have different phases. Reality or self-adjointness is essential for the reflection-probability conclusion: a complex absorbing potential need not have equal left and right reflectances. Equal asymptotic speeds are also used in identifying transmission with the squared amplitude.

Quantum tunnelling is transmission through a region where the particle's energy is below the classical potential energy. The stationary wavefunction decays within that region but need not vanish at its far boundary, so matching can leave a nonzero transmitted probability current.

This does not require a particle to borrow energy temporarily. Energy is conserved for a time-independent barrier; the classically forbidden region is described by an evanescent wave, with the same stationary energy as the incident and transmitted waves.

The gap is not a fixed mechanical separation. Thermal fluctuations of the wall compete with drift towards the microtubule under the opposing force. A monomer can be incorporated only after the gap reaches its length. The Smoluchowski equation describes this overdamped drift and diffusion of gap probability density.

For gap potential energy $U(x)=Fx$, the equation and its probability current areThe drift mobility is $D\beta$, the overdamped consequence of the fluctuation-dissipation relation for a Langevin particle. Equivalently, this coefficient is fixed by requiring the zero-current solution to be the Boltzmann distribution $p\propto e^{-\beta U}$. Solve this equation with the appropriate exclusion, incorporation and probability-normalization conditions. The steady flux through the incorporation threshold gives the addition rate $j_0$; the elongation speed is $v=\delta j_0$.

There is a regime distinction in the printed problem. The absorbing boundary imposed in part (c) means instantaneous incorporation once a gap opens, and leads to the diffusion-limited Brownian ratchet with constant load. A genuinely reaction-limited polymerization Brownian ratchet instead assumes fast gap equilibration followed by a finite chemical addition rate, giving an exponential gating factor multiplying that chemical rate. The equations requested in the subsequent parts specify the former model, which is the one solved here.

Outside a finite-range scattering potential, the stationary Schrödinger equation becomes the free Helmholtz equation. The specified incident state is a plane wave along $+z$. The scattered disturbance must be outgoing: the scattering experiment has an incoming incident beam but no independently imposed spherical wave arriving from infinity. At large radius its angular pattern is encoded by a scattering amplitude $f_k(\theta ,\varphi )$:An outgoing three-dimensional wave has amplitude proportional to $1/r$ because its probability current spreads over area proportional to $r^2$. Indeed its radial current is $(\hslash k/m)|f_k{|}^2/r^2$ to leading order, while the incident current density is $\hslash k/m$. Therefore $d\sigma /d\Omega =|f_k{|}^2$. The sign $e^{+ikr}$ assumes the stationary time factor $e^{-i{E}_{k}t/\hslash }$.

Outside a short-range scattering region, the stationary Schrödinger equation reduces to the two-dimensional Helmholtz equation. Separation in polar coordinates gives outgoing radial Hankel functions, whose large-$r$ dependence is $r^{-1/2}{e}^{ikr}$. The same power follows from conserved probability current: a circle has length proportional to $r$, so the squared outgoing amplitude must decrease as $1/r$.

For an incident plane wave in the $x$ direction, adopt the dimensionless amplitude conventionThe phase $e^{i\pi /4}$ and factor $k^{-1/2}$ specify the definition of $f$; they can be absorbed into a differently normalized scattering amplitude. Here they are kept exactly so that the later two-dimensional scattering cross-section follows consistently.

The incident plane wave carries probability current $j_{\mathrm{in}}=\hslash k/m$. For the separated outgoing wave from the previous part,The rate through a distant arc is $r{j}_r^{\mathrm{sc}}d\theta$. Dividing by incident flux defines the differential two-dimensional scattering cross-section,Because $f$ is dimensionless, $\lambda$ has units of length, as appropriate for scattering in a plane.

For a short-range potential, an incident plane wave of unit amplitude and wave vector $k{\mathbf{e}}_z$ has the large-distance scattering wavefunctionThe coefficient $f$ is the scattering amplitude. For a central potential it depends only on the scattering angle $\theta$. The incident probability current is $\hslash k/m$, while the scattered radial current is $(\hslash k/m)|f{|}^2/r^2$ to leading order. Their ratio givesThese are the differential scattering cross-section and total cross section for elastic scattering with the same incident and outgoing wave number.

The expression is asymptotic: it specifies the outgoing wave's leading $1/r$ term, not its finite-distance radial corrections or its form inside the interaction region. In particular, for angle-dependent $f$, the spherical term by itself has an angular-Laplacian contribution of order $r^{-3}$ to the free wave equation, which is cancelled by subleading terms in the exact solution.

For a positive step potential and left incidence, take $E\to V_0^{+}$ at fixed incident probability amplitude $A$. The transmitted wavenumber tends to zero while the incident wavenumber tends to $k=\sqrt{2m{V}_0}/\hslash$. Wavefunction matching at a finite potential step gives reflected amplitude $A$ and transmitted amplitude $2A$, producing the spatial wavefunction in the title. The left probability density is $4|A{|}^2{\cos }^{2}kx$ and the right density $4|A{|}^2$. The transmitted probability current still tends to zero because its velocity tends to zero. Over adjacent intervals of length $2\pi /k$, the right integrated density is twice the left. These ratios use finite local integrals of a non-normalizable stationary scattering state.

Ifthen the ratio of scattered radial probability current through $r^{2}d\Omega$ to the incident probability flux is

For equal-mass one-dimensional scattering between regions with real wavenumbers $k_0>k_2>0$, insert a layer of width $a$ and wavenumber $k_1$. For a purely right-moving incident wave of amplitude $A$ and no reflection, continuity at the first edge gives interior amplitudes $C=A(1+k_0/k_1)/2$ and $D=A(1-k_0/k_1)/2$. If $k_{1}a=\pi /2$, continuity at the second edge gives a transmitted edge amplitude $F=i(k_0/k_1)A$ and the matching condition $k_1^2=k_0{k}_2$. Thus $F/A=i\sqrt{k_0/k_2}$ and the transmitted probability current equals the incident current, despite their unequal amplitudes.

For potentials $0,V_1,V_2$ with $E>V_2>V_1>0$, the condition is $(E-V_1{)}^2=E(E-V_2)$, or $E=V_1^2/(2V_1-V_2)$, requiring $V_2/2<V_1<V_2$. It is the quantum-wave counterpart of a quarter-wave impedance transformer: the intermediate wavenumber is the geometric mean and the two reflected amplitudes cancel.

A particle incident below a semi-infinite step has a unit-modulus quantum reflection amplitude but a nontrivial phase. Continuity of the oscillating and evanescent solutions yields the displayed coefficient, for the incident phase origin at the interface. Its decaying exterior wave has zero net probability current. The phase depends on the chosen scattering convention; the unit reflection probability does not.

For equal exterior potential energies and constant mass, the exact transmission probability isBelow the barrier, the hyperbolic sine is positive for $a>0$, so $0<T<1$. At $E=V_0$, the interior wavefunction is linear in $x$; either neighbouring formula has the displayed finite limit, since $\sinh z/z$ and $\sin z/z$ tend to one. Above the barrier, the interior wavefunction oscillates: the quantum reflection probability generally remains nonzero, but $qa=n\pi$ for a positive integer $n$ gives perfect transmission probability $T=1$ through constructive wave interference. These are scattering resonances, not below-barrier quantum tunnelling.

At half height, $E=V_0/2$, we have $k=\kappa$ and $T={\mathrm{sech}}^2(\kappa a)$, consistent with finite square barrier transmission at half barrier height after replacing this width $a$ by that example's width $2a$. In the illustration, the width is four times $\ell =\hslash /\sqrt{2m{V}_0}$ and $E/V_0=1/2$, giving $T\simeq 0.01388$. The plotted wavefunction comes from solving the four interface equations; the transmission probability curve uses the closed formula.

A specified asymptotic configuration of the particles and their internal quantum states in quantum scattering. An incoming channel can scatter into several outgoing channels; S-matrix elements give their probability amplitudes with a stated probability current normalization. A channel can be energetically inaccessible even when other channels are open.

A scattering cross-section divides the scattered rate per target by the incident particle flux. Its differential form resolves that rate by outgoing direction. It has units of area for three-dimensional scattering and length for two-dimensional scattering. In two dimensions, the convention $\psi \sim e^{ikr\cos \theta }+e^{i\pi /4}f(\theta )e^{ikr}/\sqrt{kr}$ gives $d\sigma /d\theta =|f(\theta ){|}^2/k$ by comparing outgoing and incident probability currents.

The Schrödinger equation governs the evolution of a quantum state between instances of measurement in quantum mechanics. For a particle of mass $m>0$ moving in a prescribed real potential energy $V(\mathbf{r},t)$, its position wavefunction satisfiesHere $h$ is the Planck constant, ${\nabla }^2$ is the Laplace operator, and the Hamiltonian operator combines kinetic energy with potential energy. Specify an initial wavefunction, an operator domain and any boundary conditions: the differential expression alone does not completely specify a physical model. The Born rule then turns the evolving probability amplitude into position probabilities through $\rho =|\psi {|}^2$ and integration over a region.

This scalar Schrödinger operator describes nonrelativistic quantum mechanics, with relevant momenta much smaller than $mc$, where $c$ is the speed of light. It omits the constant rest energy $m{c}^2$, assumes fixed particle number, and suppresses active spin dynamics and coupling to a magnetic vector potential. A free electron with a fixed, uncoupled spin state can still use this scalar model; the Pauli equation supplies the additional spin components and magnetic coupling when needed. A prescribed real $V$ describes a closed evolution model or an approximation in which the source of that potential energy is treated as external. Dissipation, particle creation and a general measurement in quantum mechanics need further modelling.

For a time-independent self-adjoint Hamiltonian operator, the spectral theorem defines unitary time evolution byInitial states in the operator domain satisfy the differential equation in the Hilbert space sense; the unitary operator also evolves all other states in that Hilbert space. Time-dependent Hamiltonian operators need suitable regularity and domain hypotheses; their evolution is not generally the exponential of $-itH(t)/\hslash$.

The local reason for conservation is especially tangible. Multiply the Schrödinger equation by the complex conjugate ${\psi }^{\ast }$, subtract its conjugate multiplied by $\psi$, and use the fact that the real potential energy terms cancel. For smooth wavefunctions this gives the probability continuity equationThe divergence theorem identifies the change in probability inside a region with minus the outward probability current. Decay at infinity, or appropriate probability-conserving boundary conditions, preserves wavefunction normalization. Conservation of total probability does not require a stationary probability density.

Its usefulness comes from being a linear partial differential equation: quantum superpositions of solutions are solutions, while different quantum phases can produce interference. For a fixed Hamiltonian operator, the time-independent Schrödinger equation finds the energy eigenstates; each then acquires just its own time-dependent quantum phase. One eigenvalue problem therefore organizes both allowed energies and the evolution of arbitrary quantum superpositions, with a continuous spectrum included when required. This is the attractive bridge between a spatial wave problem and a clock: energy differences determine the rate at which relative quantum phases change. It makes atomic spectral lines, quantum tunnelling and quantum scattering calculable from a specified model.

For an explicit example without confinement, take a free particle on the whole line, $V=0$, initially in a Gaussian wave packet with position standard deviation ${\sigma }_0$ and mean momentum $\hslash k_0$:Unlike a single plane wave, this wavefunction satisfies wavefunction normalization. A Fourier transform writes it as a quantum superposition of plane waves. Their dispersion relation is $\omega (k)=\hslash k^2/(2m)$, so each Fourier component gains the factor $e^{-i\hslash k^{2}t/(2m)}$. Completing the square in the resulting Gaussian integral giveswith the square root chosen continuously from $t=0$. Its position probability density and standard deviation areThe centre moves with the group velocity ${\omega }^{\prime }(k_0)=v_0$ and the wave packet spreads because different wavenumbers have different group velocities. This is unitary wave dispersion, without a random force or loss of probability. Independently, the initial momentum standard deviation is $\Delta p=\hslash /(2{\sigma }_0)$ and the initial position-momentum covariance vanishes. Free evolution gives $x(t)=x(0)+p(0)t/m$ in the Heisenberg picture, henceThis independently checks the spreading law. The Gaussian integral of $\rho$ is one at every time; the decreasing peak is exactly offset by increasing width.

As hypothetical preparation parameters, choose a free electron with ${\sigma }_0=1\mathrm{nm}$ and $k_0=2{\mathrm{nm}}^{-1}$. Using $\hslash \simeq 1.054571817\times 10^{-34}\mathrm{J}\mathrm{s}$ and $m\simeq 9.1093837\times 10^{-31}\mathrm{kg}$ givesThe rounded electron mass uses the 2022 adjustment.^[ref] At $t=t_s$, the centre is $4\mathrm{nm}$ and ${\sigma }_t=\sqrt{2}\mathrm{nm}$; at $t=2t_s$, they are $8\mathrm{nm}$ and $\sqrt{5}\mathrm{nm}$. The mean kinetic energy staysIt includes both mean motion and the momentum spread. The mean speed is about $7.72\times 10^{-4}c$ and the momentum spread corresponds to $1.93\times 10^{-4}c$, making the nonrelativistic model accurate for the appreciably occupied part of this ideal Gaussian distribution. Its mathematically unbounded tails are an idealization.

The complex width also explains where the probability goes. Write $\psi =\sqrt{\rho }{e}^{i\theta }$, where $\theta$ is the dimensionless quantum phase. Differentiating the exact solution givesRelative to the moving centre, the probability current $j_{\mathrm{rel}}=j-v_0\rho$ points outward on both sides for $t>0$. This is a flux statement, not an additional assumption about individual particle trajectories. At $t=t_s$ and $x-v_{0}t=\pm {\sigma }_0$, the relative flow speeds $j_{\mathrm{rel}}/\rho$ are $\pm {\sigma }_0/(2t_s)$, about $\pm 2.8942\times 10^4\mathrm{m}{\mathrm{s}}^{-1}$. The probability within a moving interval of half-width ${\sigma }_0$ falls from about $0.6827$ initially to $0.5205$ at $t_s$, exactly because of this outward probability current.

Historically, the 1926 Quantisierung als Eigenwertproblem (Erste Mitteilung) by Erwin Schrödinger began with the stationary hydrogen atom problem. It distinguishes discrete negative energies from a positive continuous spectrum; it is not yet the general evolution equation displayed above.^[ref] The opening of quantum mechanics discusses its original vibrating-string analogy.

In the 12 December 1933 lecture The fundamental idea of wave mechanics by Erwin Schrödinger, Erwin Schrödinger explains the move from independent rays to waves through diffraction, then applies that analogy to atomic physics. His closing pages explicitly acknowledge the difficulty of reconciling wave language with observed particle tracks. This retrospective physical argument motivates the model; it does not derive the modern Born rule from classical mechanics.^[ref]

The dates separate proposal from measurement: the de Broglie wavelength belongs to the 1924 thesis, the Schrödinger equation to 1926, and the Davisson–Germer experiment's electron diffraction beams to 1927.^[ref] That linked experiment describes the prepared nickel crystal, evacuated electron gun, angular collector scans, retarding-potential selection and heating controls. Its measured collector electric current tests wave-dependent scattering probabilities; it neither measures a wavefunction directly nor establishes this equation uniquely.

For overdamped Brownian motion at uniform temperature in a potential energy $U$, constant diffusion coefficient $D$ gives the displayed Fokker-Planck equation, with $\beta =(k_{B}T{)}^{-1}$. The probability current is $\mathbf{J}=-D(\nabla p+\beta p\nabla U)$. Zero current recovers the Boltzmann distribution; absorbing boundaries instead permit transport and mean first-passage times. This diffusion equation should be distinguished from the similarly named coagulation equation.

Where a nonzero wavefunction can be written as $\psi =\sqrt{\rho }{e}^{i\theta }$, and there is no vector potential, substituting into the Schrodinger probability current gives $\mathbf{j}=\rho \hslash \nabla \theta /m$. The direction of probability flow is determined by the phase gradient rather than the slope of the real part. A plane wave and its complex conjugate have the same real part and density but opposite currents. In a vector potential the gauge-invariant kinetic momentum replaces $\hslash \nabla \theta$.

For constant mass $m$, a finite jump from potential energy zero at $x<0$ to $V_0>0$ at $x>0$ requires continuity of the wavefunction and its first derivative. With incidence only from the left and $E>V_0$, put $k=\sqrt{2mE}/\hslash$ and $q=\sqrt{2m(E-V_0)}/\hslash$. Matching $A+B=C$ and $k(A-B)=qC$ gives reflected probability amplitude $B/A=(k-q)/(k+q)$ and transmission amplitude $C/A=2k/(k+q)$. The transmission probability is $(q/k)|C/A{|}^2$, conserving probability current.

For $0<E<V_0$, replace $q$ by $i\kappa$, where $\kappa =\sqrt{2m(V_0-E)}/\hslash$ is the tunnelling decay constant. The right-hand wavefunction is $C{e}^{-\kappa x}$, with zero probability current. Its probability density decays on length $1/(2\kappa )$, while its amplitude decays on length $1/\kappa$. The semi-infinite raised region gives total reflection, unlike a finite potential barrier allowing quantum tunnelling into a second allowed region.

For a time-independent Hamiltonian operator $H$, an energy eigenstate satisfiesFor a one-dimensional particle of mass $m$ in a real potential energy $U(x)$,Its full time-dependent wavefunction is $\Psi (x,t)=\psi (x)e^{-iEt/\hslash }$: substitution in the Schrödinger equation checks both the minus sign and the factor $\hslash$. Thus an energy eigenstate has stationary position probability density, even though its quantum phase evolves. The boundary conditions and operator domain are part of this eigenvalue problem, selecting physically allowed solutions; the infinite square well and Radial Schrodinger equation for the hydrogen atom give complementary confined examples.

For a complete discrete orthonormal basis of energy eigenstates, linearity gives $\Psi (x,t)={\sum }_n{c}_n{\psi }_n(x)e^{-i{E}_{n}t/\hslash }$, with ${\sum }_n|c_n{|}^2=1$. Cross terms in $|\Psi {|}^2$ oscillate at angular frequencies $(E_n-E_m)/\hslash$. A continuous spectrum requires the corresponding spectral integrals; for a free particle on the whole line, individual plane waves are generalized eigenfunctions, and a normalized Gaussian wave packet is their integral quantum superposition. Adding a spatially constant real $C$ to $U$ shifts every energy eigenvalue by $C$ and multiplies the full state by $e^{-iCt/\hslash }$, a global phase; energy differences, position probabilities and probability current are unchanged.