Quantum That ClicksEpisode 4 / The Schrödinger Equation and Quantization

The equation of motion for the wave ψ ── and why "energy in discrete steps" is the plain consequence of trapping a continuous wave in a box

The Schrödinger Equation and Quantization The rule that tells you how ψ changes moment to moment is the Schrödinger equation.
Trap a wave in a box and only standing waves with nodes at both ends are allowed ── so, just like the overtones of a string, energy comes in discrete steps.

Tools you'll need: Episode 1's p=h/λ, Episode 2's ψ and phase, standing waves on a string This episode: iℏ ∂ψ/∂t = Ĥψ, E_n ∝ n²

In Episodes 2 and 3 we looked at the wave function $\psi$ and at adding up its phases $S/\hbar$. So how, exactly, does $\psi$ change shape from moment to moment? The "equation of motion for the wave" that gives this directly is the Schrödinger equation $i\hbar\,\partial\psi/\partial t=\hat H\psi$. And when you trap this wave in a box, quantum theory's most famous face appears ── energy comes in "discrete steps" (quantization). But there's something we want to stress here. Things become discrete not because the world is discrete. It is exactly the same as a guitar string only sounding at particular pitches (overtones) ── confine a continuous wave with both ends fixed, and only the waves that "fit" get selected, and as a result you get discrete steps. Quantization is a perfectly ordinary consequence of being a wave.

01We need an equation of motion for the wave ── Schrödinger

If $\psi$ is a wave, there must be an equation deciding how the wave spreads. Writing that down by translating the energy relation "kinetic energy + potential energy = total energy" into the language of waves is the Schrödinger equation.

The Schrödinger equation ── the time evolution of ψ

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

On the left, $i\hbar\,\partial_t$ represents "energy"; on the right, $\hat H$ (the Hamiltonian) represents the kinetic + potential energy. The presence of $i$ and $\hbar$ is the crux ── thanks to $i$, the solution becomes an oscillating wave, and the phase spins at the rate $E/\hbar$ (exactly Episode 1's $E=\hbar\omega$). The $i$ left over in Episode 3 of "Cosmology That Clicks" is this $i$.

02Stationary states ── waves whose phase alone spins

As a special solution, there are waves whose shape doesn't change and whose phase alone spins. This is a state of definite energy ($\hat H\psi=E\psi$), and its time dependence is $\psi(t)=\psi\,e^{-iEt/\hbar}$ ── since the magnitude $|\psi|^2$ is constant in time, we call it a stationary state. The rate at which the phase spins is $E/\hbar=\omega$. The higher the state's energy, the faster the phase spins. The "orbits" of an electron in an atom are these stationary states. So can that energy $E$ take any value at all? ── Confine it, and it cannot.

03Trap it in a box and it becomes discrete

The simplest confinement ── put an electron in a box of width $L$ (walls on both sides). Since it can't leave the walls, $\psi$ must be zero at both ends. This is exactly the same condition as a string fixed at both ends. So the only waves allowed are standing waves with nodes at both ends ── waves where an integer number of half-wavelengths fits exactly.

Let's try it ── wavelengths that fit → discrete energies

If n half-wavelengths fit exactly, the wavelength is

$$\lambda_n=\frac{2L}{n}\quad(n=1,2,3,\dots)$$

Using de Broglie's p=h/λ (Episode 1) and E=p²/2m gives

$$E_n=\frac{p_n^2}{2m}=\frac{n^2 h^2}{8mL^2}$$

Energy can only take the discrete values ($E_n\propto n^2$) for $n=1,2,3\dots$. Even though the wave is continuous, the boundary condition of confinement selects out the allowed waves by an integer $n$ ── that is quantization. It is precisely the same logic as a guitar string only sounding the fundamental, second harmonic, third harmonic, and so on.

This episode's quantum number ── the integer n

The $n$ (quantum number) that counts the discrete steps is a unitless integer. Like the inverse-square power $d-1$ in Episode 5 ("Fields That Click"), or the Yukawa coupling of mass, a physical "count" always shows up as a dimensionless integer or ratio. $n$ is literally "how many half-wavelengths fit in the box."

04Give it a spin ── the wave inside the box

The figure below. In a box of width $L$, we draw the wave $\psi=\sin(n\pi x/L)$. Use the slider to move $n$ continuously. Only when $n$ is an integer does the right end of the wave land exactly at the wall as a node (zero), so it "fits" (green = allowed). Stray from an integer and the right end sticks out past the wall, so it doesn't fit (red = not allowed). The ladder on the right shows the allowed energies $E_n\propto n^2$.

Even as you turn $n$ continuously, the only physically allowed values are the rungs of the ladder (integer $n$). On top of a world of continuous waves, the boundary condition applies a "discrete sieve" ── this is the true identity of quantization. Shrink the box (small $L$) and the spacing of the rungs ($\propto1/L^2$) widens, making the quantization stand out. Confine it as small as an atom and the discreteness of the energy levels becomes conspicuous, producing light of fixed colors specific to each atom (spectral lines).

Figure: the wave sin(nπx/L) inside a box of width L. Move n continuously and only at integers does the right end land as a node at the wall and "fit" (green = allowed / red = not allowed). The ladder on the right = the allowed energies E_n∝n²

n (number of half-wavelengths that fit in the box) 2.00

fitting wave (allowed, integer n) non-fitting wave (not allowed)

05Quantization is not a "discrete world" ── a continuous wave, selected at the boundary

This is what we most want to convey in this episode. The impression that "quantum = the world is discrete (digital)" is not accurate. The Schrödinger equation is the equation of a smooth, continuous wave. Space and time are both continuous, and $\psi$ varies continuously. Things become discrete only when the wave is confined, because only the fitting modes get selected ── on a continuous foundation, boundary conditions produce discreteness. A free electron with no confinement can take energy continuously (it does not become discrete). Discreteness is not the discreteness of the world, but the consequence of wave + confinement.

Continuous or discrete ── a record and a CD are the same song We tend to contrast "quantum is discrete, relativity is continuous," but what this episode shows is subtler. The quantum foundation (the wave $\psi$) is continuous, and discreteness is something emergent from confinement. A continuous wave and its discrete modes (overtones) are two views of the same single string ── just as a record (a continuous waveform) and a CD (discrete numbers) carry the same song, neither is more "real." In fact, the mathematics that goes back and forth between a smooth waveform and its discrete components (spectrum) is the Fourier transform, and quantum theory stands on top of it (position and momentum, time and energy, are the front and back of a Fourier pair mediated by $\hbar$). Episode 5's uncertainty comes precisely from this two-sidedness of "continuous ↔ discrete / spread ↔ wavelength." It is a theme continuous with the sampling in Episode 5 of "Cosmology That Clicks."

◇ ◇ ◇

The honest line ── the reach of the Schrödinger equation

The Schrödinger equation $i\hbar\partial_t\psi=\hat H\psi$, the stationary state $\psi e^{-iEt/\hbar}$, the quantization of a particle in a box $E_n=n^2h^2/8mL^2$ and $\lambda_n=2L/n$, the continuous spectrum of a free particle, the energy levels and emission spectra of atoms, and the fact that quantization comes from the standing-wave boundary conditions (isomorphic to a string's overtones) are all established physics.

However. ① The Schrödinger equation is non-relativistic ($\beta\ll1$; it doesn't include Episode 1's "story of c"). To reconcile relativity and quantum theory you need quantum field theory (the Dirac equation and so on, SR + QM = "Fields That Click") ── relativity and quantum theory are not "in opposition"; they merge here. ② A real atom is confined not by a box but by the nucleus's Coulomb attraction (in 3D), and its quantum numbers include not just $n$ but angular momentum and others. The "box" is the simplest model. ③ "Quantization = wave + confinement" is a story about bound states; there is also discreteness, such as spin, that isn't easily reduced to a "wave in space." ④ Whether spacetime itself is discrete or continuous at the Planck scale is unsettled (quantum gravity) ── that is not this episode's topic.

Practice problems (solvable with just this episode's formulas)

For a particle in a box, what is the ratio of the energies of n=1 and n=2? (E_n∝n²)
Show the answer
$E_2/E_1=2^2/1^2=4$. Four times the ground state (n=1). The spacing of the steps widens as n².
If you halve the box width L, what happens to the energy levels E_n∝1/L²? And the stronger the confinement?
Show the answer
$1/(1/2)^2=4$ times. The stronger the confinement (small L), the larger both the energy and the spacing of the steps, and the more pronounced the quantization. This is why it becomes conspicuous at atomic sizes.
Why are waves allowed only at integer n? Answer via the common ground with a string's overtones.
Show the answer
Since both ends are walls (nodes), only waves in which an integer number of half-wavelengths fit exactly (nodes at both ends) are allowed. It's the same as a string fixed at both ends only sounding the fundamental and its overtones. The boundary condition selects the integer n.
Why is "quantum = the world is discrete" not accurate?
Show the answer
The Schrödinger equation is the equation of a continuous wave, and space, time, and ψ are continuous. Discreteness is an emergent result of only the fitting modes being selected when a wave is confined. A free particle has a continuous spectrum. Discreteness is a consequence of "wave + confinement," not the discreteness of the world.

Episode 4 summaryQuantization is the shadow of a continuous wave trapped in a box

The equation of motion for the wave function $\psi$ is the Schrödinger equation $i\hbar\partial_t\psi=\hat H\psi$. Thanks to $i$, the solution is an oscillating wave, and a stationary state has its phase spinning at $E/\hbar=\omega$ (Episode 1's $E=\hbar\omega$). Trap this wave in a box of width $L$ and only standing waves with nodes at both ends are allowed; only waves with an integer $n$ half-wavelengths fitting get selected, and energy becomes discrete, $E_n=n^2h^2/8mL^2$ ── the same as a guitar's overtones. The quantum number $n$ is a unitless integer.

The key point is that discreteness (quantization) is not because "the world is discrete." The Schrödinger equation is the equation of a continuous wave, and discreteness is an emergent consequence of the fitting modes being selected when a wave is confined (a free particle is continuous). A continuous waveform and its discrete modes are, like a record and a CD, two views of the same thing, and what connects them is the Fourier transform ── the stage for next episode's uncertainty. "Continuous or discrete" is a matter of description, not of the foundation; what matters is still $\hbar$.

Coming up ── Episode 5 "Uncertainty ── the smallest area that ℏ spans." Position and momentum cannot both be pinned down exactly at the same time: $\Delta x\cdot\Delta p\ge\hbar/2$. This is not clumsy measurement, but comes from a general property of waves (the two sides of a Fourier pair): a wave cannot simultaneously "gather in a narrow spot" and "have a definite wavelength (= momentum)." ── the same way a short sound has an ambiguous pitch and a long, pure tone has an ambiguous position (moment). In the space of position and momentum, there is a grain of smallest area called $\hbar$. Into the heart of the story that connects continuous and discrete.

This document is Episode 4 of the "Quantum That Clicks" series, a piece of reading for physics-loving high-schoolers and undergraduates. The Schrödinger equation $i\hbar\partial_t\psi=\hat H\psi$, the stationary state $\psi(t)=\psi\,e^{-iEt/\hbar}$ ($\hat H\psi=E\psi$), the eigen-energies of the infinite well (box) $E_n=n^2h^2/8mL^2$ and wavelengths $\lambda_n=2L/n$, the continuous spectrum of a free particle, the quantization from boundary conditions (standing waves) and its isomorphism with a string's overtones, and the discrete atomic levels and emission lines are all standard physics. That the Schrödinger equation is non-relativistic and that reconciling it with SR requires quantum field theory (the Dirac equation, etc.) (relativity and quantum theory are not in opposition but merge in QFT), that a real atom is a 3D Coulomb-bound problem with several quantum numbers, that there is hard-to-reduce discreteness such as spin, and that the discreteness/continuity of spacetime at the Planck scale is unsettled (quantum gravity) are all noted in the body's "honest line." The figure is a schematic of the standing waves and discrete levels of the infinite well. ── To print, use your browser's "Print" and "Save as PDF" (in the print version the slider and answers are static and hidden). Adjacent episodes: Episode 3: Summing over all paths / Contents / sister series Cosmology That Clicks (sampling).

Print / make a PDF: ⌘+P (on Windows, Ctrl+P). On screen, moving n makes the wave "fit" the box only at integers (green). Click "Show the answer" to open a solution.

The Schrödinger Equation and Quantization The rule that tells you how ψ changes moment to moment is the Schrödinger equation.Trap a wave in a box and only standing waves with nodes at both ends are allowed ── so, just like the overtones of a string, energy comes in discrete steps.

01We need an equation of motion for the wave ── Schrödinger

02Stationary states ── waves whose phase alone spins

03Trap it in a box and it becomes discrete

04Give it a spin ── the wave inside the box

05Quantization is not a "discrete world" ── a continuous wave, selected at the boundary

Episode 4 summaryQuantization is the shadow of a continuous wave trapped in a box

The Schrödinger Equation and Quantization The rule that tells you how ψ changes moment to moment is the Schrödinger equation.
Trap a wave in a box and only standing waves with nodes at both ends are allowed ── so, just like the overtones of a string, energy comes in discrete steps.