40.1.1 Justifying the Schrödinger Equation

The Schrödinger equation can be neither derived nor proved. It is not an outgrowth of any previous theory. Its success depended on its ability to explain the various phenomena that had refused to yield to a classical-physics analysis and to make new predictions that were subsequently verified.

Although the Schrödinger equation cannot be derived, the reasoning behind it can at least be made plausible. De Broglie had postulated a wave-like nature for matter in which a particle of mass $m$ , velocity $v$ , and momentum $p = m v$ has a wavelength

λ = \frac{h}{p} = \frac{h}{m v}

(40.2)

Schrödinger’s goal was to find a wave equation for which the solution would be a wave function having the de Broglie wavelength.

An oscillatory wave-like function with wavelength $λ$ is

ψ (x) = ψ_{0} sin (\frac{2 π x}{λ})

(40.3)

where $ψ_{0}$ is the amplitude of the wave function. Suppose we take a second derivative of $ψ (x) :$

\frac{d^{2} ψ}{d x^{2}} = \frac{d}{d x} \frac{d ψ}{d x} = \frac{d}{d x} [\frac{2 π}{λ} ψ_{0} cos (\frac{2 π x}{λ})] = - \frac{{(2 π)}^{2}}{λ^{2}} ψ_{0} sin (\frac{2 π x}{λ})

We can use Equation 40.3 to write this as

\frac{d^{2} ψ}{d x^{2}} = - \frac{{(2 π)}^{2}}{λ^{2}} ψ (x)

(40.4)

Equation 40.4 relates the wavelength $λ$ to a combination of the wave function $ψ (x)$ and its second derivative.

NOTE These manipulations are not specific to quantum mechanics. Equation 40.4 is well known for classical waves, such as sound waves and waves on a string.

Schrödinger’s insight was to identify $λ$ with the de Broglie wavelength of a particle. We can write the de Broglie wavelength in terms of the particle’s kinetic energy $K$ as

λ = \frac{h}{m v} = \frac{h}{\sqrt{2 m (\frac{1}{2} m v^{2})}} = \frac{h}{\sqrt{2 m K}}

(40.5)

Notice that the de Broglie wavelength increases as the particle’s kinetic energy decreases. This observation will play a key role.

If we square this expression for $λ$ and substitute it into Equation 40.4, we find

\frac{d^{2} ψ}{d x^{2}} = - \frac{{(2 π)}^{2} 2 m K}{h^{2}} ψ (x) = \frac{- 2 m}{ℏ^{2}} K ψ (x)

(40.6)

where $ℏ = h / 2 π$ . Equation 40.6 is a differential equation for the function $ψ (x)$ . The solution to this equation is the sinusoidal wave function of Equation 40.3, where $λ$ is the de Broglie wavelength for a particle with kinetic energy $K$ .

Our derivation of Equation 40.6 assumed that the particle’s kinetic energy $K$ is constant. The energy diagram of FIGURE 40.1a reminds you that a particle’s kinetic energy remains constant as it moves along the $x$ -axis only if its potential energy $U$ is constant. In this case, the de Broglie wavelength is the same at all positions.

The figure labeled (“a”) consists of two parts that illustrate the energy diagram and the wave function of a particle whose de Broglie wavelength is constant.

The figure labeled (b) consists of two parts that illustrate the energy diagram and the wave function of a particle whose de Broglie wavelength is varying.

Full Alternative Text

In contrast, FIGURE 40.1b shows the energy diagram for a particle whose kinetic energy is not constant. This particle speeds up or slows down as it moves along the $x$ -axis, transforming potential energy into kinetic energy or vice versa. Consequently, its de Broglie wavelength changes with position.

Suppose a particle’s potential energy—gravitational or electric or any other kind of potential energy—is described by the function $U (x)$ . That is, the potential energy is a function of position along the axis of motion. For example, the potential energy of a mass on a spring is $\frac{1}{2} k x^{2}$ .

If $E$ is the particle’s total mechanical energy, its kinetic energy at position $x$ is

K = E - U (x)

(40.7)

If we use this expression for $K$ in Equation 40.6, that equation becomes

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

This is Equation 40.1, the Schrödinger equation for the particle’s wave function $ψ (x)$ .

NOTE This has not been a derivation of the Schrödinger equation. We’ve made a plausibility argument, based on de Broglie’s hypothesis about matter waves, but only experimental evidence will show if this equation has merit.