The Schrödinger Equation in One Dimension

The de Broglie hypothesis assigns a wavelength $λ = h / p$ to every particle, and the two-slit experiment shows that electrons interfere. A wave that interferes must obey a wave equation. The equation governing the matter wave of a nonrelativistic particle was found by Erwin Schrödinger late in 1925, and it plays the role for atomic, molecular, and solid-state physics that Newton's second law plays for classical mechanics.¹

The Schrödinger equation is not derived. Like Newton's laws, it is a fundamental postulate, and its authority rests entirely on agreement with experiment. What can be derived is the form the equation must take, by requiring that it reproduce the de Broglie relations for a free particle and that it be linear in the wave function. Schrödinger's equation is nonrelativistic; the relativistic wave equation for the electron was supplied by Dirac in 1928. Within the nonrelativistic range it is exact for practical purposes.

The form dictated by de Broglie

Start from the classical wave equation for the electric field of light, which propagates at $c$ ,

\frac{\partial ^{2} E}{\partial x ^{2}} = \frac{1}{c ^{2}} \frac{\partial ^{2} E}{\partial t ^{2}} .

A harmonic solution $E (x, t) = E_{0} cos (k x - ω t)$ differentiated twice in space brings down $- k^{2}$ and twice in time brings down $- ω^{2}$ , so substitution gives $k^{2} = ω^{2} / c^{2}$ , i.e. $ω = k c$ . Using $ω = E /ℏ$ and $p = ℏ k$ this reads $E = p c$ , the correct energy-momentum relation for a photon. The classical wave equation encodes the photon dispersion because it relates a second space derivative to a second time derivative.

For a massive particle the dispersion is different. The nonrelativistic energy is

E = \frac{p ^{2}}{2 m} + V,

and substituting the de Broglie relations $E = ℏ ω$ , $p = ℏ k$ gives

ℏ ω = \frac{ℏ ^{2} k ^{2}}{2 m} + V .

This differs from the photon relation in two ways: it carries the potential energy $V$ , and $ω$ is quadratic in $k$ rather than linear. Reading off the powers, one factor of $ω$ comes from a first time derivative and the factor $k^{2}$ from a second space derivative. The wave equation for matter must therefore relate the first time derivative to the second space derivative and must contain $V$ — the opposite balance of derivatives from the classical wave equation.²

A third requirement is linearity: every term must be linear in the wave function, so that any linear combination of solutions for the same potential is again a solution. Linearity is what lets matter waves add constructively and destructively, the property interference demands.

The time-dependent equation

The equation meeting these requirements, postulated for a particle of mass $m$ in one dimension, is the time-dependent Schrödinger equation:

- \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2} Ψ ( x , t )}{\partial x ^{2}} + V (x, t) Ψ (x, t) = i ℏ \frac{\partial Ψ ( x , t )}{\partial t} .

The explicit factor $i = - 1$ is the sharpest break from the classical wave equation. A real cosine cannot solve it: differentiating once in time turns a cosine into a sine, while the second space derivative returns a cosine, so the two sides cannot match. The exponential form does work. For a free particle, $V = V_{0}$ constant, try

Ψ (x, t) = A e^{i (k x - ω t)} = A [cos (k x - ω t) + i sin (k x - ω t)] .

Then $\partial_{t} Ψ = - iω Ψ$ and $\partial_{xx} Ψ = - k^{2} Ψ$ , and substitution reproduces exactly $ℏ ω = ℏ^{2} k^{2} /2 m + V_{0}$ . The free-particle solution is intrinsically complex; there is no real wave function behind it.

Born's probability interpretation

Because $Ψ$ is complex it is not itself a measurable field the way the classical $E (x, t)$ is — every measurement returns a real number. What is measurable is the probability of finding the particle, and Max Born identified the rule connecting $Ψ$ to that probability.³

Physical meaning attaches not to $Ψ$ but to the product $Ψ^{*} Ψ = ∣Ψ ∣^{2}$ . The question what is waving? has no answer; $Ψ$ is a computational device whose squared modulus is a probability distribution. This is the same reading of the wave function introduced through wave packets, now attached to the equation that produces $Ψ$ .

Since the particle is certainly somewhere, the probabilities over all $x$ sum to one:

- \infty \int \infty Ψ^{*} Ψ d x = 1.

The normalization requirement is not cosmetic. Together with boundary conditions at finite $x$ , it is what restricts the allowed solutions and forces the energy of a bound particle to take discrete values, a result worked out in full for the square well.

Separating time from space

Schrödinger's own applications — the hydrogen atom and the harmonic oscillator — have potentials that do not depend on time, $V = V (x)$ . For such potentials the space and time dependence of $Ψ$ separate, and the equation collapses to something far easier to solve. Assume a product form

Ψ (x, t) = ψ (x) ϕ (t) .

Substituting into the time-dependent equation and dividing through by $ψ (x) ϕ (t)$ gives

- \frac{ℏ ^{2}}{2 m} \frac{1}{ψ ( x )} \frac{d ^{2} ψ ( x )}{d x ^{2}} + V (x) = i ℏ \frac{1}{ϕ ( t )} \frac{d ϕ ( t )}{d t} .

The left side depends only on $x$ and the right side only on $t$ . Two functions of independent variables can be equal for all $x$ and $t$ only if both equal the same constant. Call it the separation constant $C$ . The single partial differential equation splits into two ordinary ones:

- \frac{ℏ ^{2}}{2 m} \frac{1}{ψ ( x )} \frac{d ^{2} ψ ( x )}{d x ^{2}} + V (x) = C, i ℏ \frac{1}{ϕ ( t )} \frac{d ϕ ( t )}{d t} = C .

Separation of variables factors the space-time wave function into a spatial shape fixed by V(x) and a universal phase that rotates at frequency E over hbar; the squared modulus of the phase is 1.

The time equation carries no potential, so it is solved once and for all. Writing $d ϕ / ϕ = (C / i ℏ) d t = - (i C /ℏ) d t$ and integrating,

ϕ (t) = e^{- i C t /ℏ} .

This oscillates at frequency $f = C / h$ . But de Broglie fixes the frequency of the wave to $f = E / h$ , so the separation constant is the total energy,

C = E, ϕ (t) = e^{- i E t /ℏ} .

The time-independent equation

With $C = E$ , multiplying the spatial equation through by $ψ (x)$ yields the time-independent Schrödinger equation:

- \frac{ℏ ^{2}}{2 m} \frac{d ^{2} ψ ( x )}{d x ^{2}} + V (x) ψ (x) = E ψ (x) .

The full wave function of a definite-energy state is

Ψ (x, t) = ψ (x) e^{- i E t /ℏ} .

The time dependence is a pure phase of unit modulus. Its probability density is therefore static:

∣Ψ (x, t) ∣^{2} = ψ^{*} (x) e^{+ i E t /ℏ} ψ (x) e^{- i E t /ℏ} = ∣ ψ (x) ∣^{2},

independent of time. A state of definite energy is a stationary state: the phase $e^{- i E t /ℏ}$ rotates in the complex plane, but nothing observable changes.

In a stationary state the complex amplitude at a point traces a circle in the complex plane at angular rate E over hbar, while its squared length, the probability density, stays fixed.

The normalization condition inherits the same simplification. Because the phase cancels in $Ψ^{*} Ψ$ ,

- \infty \int \infty ψ^{*} (x) ψ (x) d x = 1,

a condition on the spatial factor alone.

Effect of the potential on the wave function

Rearranged, the time-independent equation reads

\frac{d ^{2} ψ}{d x ^{2}} = \frac{2 m}{ℏ ^{2}} [V (x) - E] ψ (x) .

The second derivative is the curvature of $ψ$ . Its sign relative to $ψ$ is fixed by whether the total energy exceeds the potential:

$E > V (x)$ (classically allowed). The bracket is negative, so $ψ^{''}$ and $ψ$ have opposite signs. Wherever $ψ$ is positive it curves down toward the axis, and wherever it is negative it curves up toward the axis. The wave function oscillates, like a sine or cosine. Larger $E - V$ means larger curvature, hence shorter wavelength — the classical statement that a faster particle has a shorter de Broglie wavelength.
$E < V (x)$ (classically forbidden). The bracket is positive, so $ψ^{''}$ and $ψ$ share a sign. A positive $ψ$ curves up, away from the axis; a negative $ψ$ curves down, away from the axis. The wave function grows or decays exponentially rather than oscillating.

The sign of E minus V fixes the curvature: where the energy exceeds the potential the wave function bends toward the axis and oscillates; where the potential exceeds the energy it bends away and runs off exponentially.

This curvature rule is the qualitative engine behind every problem in the chapter. In a bound-state problem the particle is allowed in a central region and forbidden outside it. Inside, $ψ$ oscillates; outside, it must decay. Only for special values of $E$ do the oscillating interior and the decaying exterior join smoothly with a solution that also vanishes at infinity. Those special energies are the quantized levels.

Acceptable wave functions

The form of $ψ (x)$ depends on $V (x)$ , but every admissible solution obeys the same regularity conditions. Where the potential jumps between regions, one solves the equation separately in each region and joins the pieces.

Because the probability density cannot change discontinuously from point to point, $ψ$ must be continuous. Because the equation involves $ψ^{''} = d^{2} ψ / d x^{2}$ , the first derivative $ψ^{'}$ (the slope) must also be continuous, so the graph of $ψ$ is smooth. The one exception is where $V \to \infty$ : no particle can have infinite potential energy, so $ψ = 0$ there, and at the boundary of such a region $ψ^{'}$ may be discontinuous.

A continuous wave function joining two regions smoothly, versus rejected candidates with a kink in the slope or a jump in value; the physics admits only the smooth match except at an infinite wall.

Collecting the requirements gives the checklist every wave function must pass.⁴

Conditions 3 and 4 exist because measurable quantities such as position and momentum are always finite and single-valued; if $ψ$ or $ψ^{'}$ failed either, the predictions built from them would not be. Condition 5 is normalizability. Together these conditions are the mathematical content of bound state: a particle trapped in a region must have a wave function that decays to nothing far away, and only special energies produce such a function.

A free-particle solution

For a free particle $V (x) = 0$ , so the time-independent equation is

- \frac{ℏ ^{2}}{2 m} \frac{d ^{2} ψ}{d x ^{2}} = E ψ ⟹ \frac{d ^{2} ψ}{d x ^{2}} = - k^{2} ψ, k^{2} = \frac{2 m E}{ℏ ^{2}} .

Any combination $ψ (x) = A sin k x + B cos k x$ solves this. Differentiating twice,

\frac{d ^{2} ψ}{d x ^{2}} = - k^{2} A sin k x - k^{2} B cos k x = - k^{2} ψ (x),

and since $ℏ^{2} k^{2} = 2 m E$ , substitution gives $E ψ = E ψ$ , an identity, so the combination is a solution for any $A$ , $B$ .⁵ With no boundary to confine it, $k$ is unrestricted and $E = ℏ^{2} k^{2} /2 m$ takes any nonnegative value: the free particle has a continuous energy spectrum. Confinement is what changes this. Impose a boundary and only discrete $k$ survive, which is the subject of the infinite and finite square wells. The continuum itself, and how genuinely localized states are assembled from it, is taken up next for the free particle and its wave packets.

Tipler & Llewellyn, Modern Physics, §6-1 — the Schrödinger equation is postulated, not derived, and validated by experiment; nonrelativistic, superseded at high speed by the Dirac equation. ↩
Tipler & Llewellyn, Modern Physics, §6-1 — building the equation from the photon dispersion and the de Broglie relations, showing it must relate the first time derivative to the second space derivative and carry $V$ . ↩
Tipler & Llewellyn, Modern Physics, §6-1 — Born's probabilistic interpretation of the complex wave function, $P (x, t) d x = Ψ^{*} Ψ d x = ∣Ψ ∣^{2} d x$ , and the normalization condition. ↩
Tipler & Llewellyn, Modern Physics, §6-1 — the five conditions for an acceptable wave function and the continuity of $ψ$ and $d ψ / d x$ at a potential discontinuity. ↩
Tipler & Llewellyn, Modern Physics, §6-1, Example 6-1 — verification that $A sin k x + B cos k x$ solves the free-particle time-independent equation. ↩