AN INFORMAL DERIVATION OF THE SCHRÖDINGER EQUATION

TERENCE TAO

Abstract. We briefly review the standard physical derivation of the free Schrödinger equation. This is a highly non-rigorous treatment and is intended only to motivate the Schrödinger equation formalism.

1. Basic derivation

In classical Newtonian mechanics, in the absence of potentials, the energy E, the momentum p⃗, the mass m, and the velocity v⃗ of a free particle are related by the formulae

E = \frac{1}{2}m|v⃗|^{2}; p⃗ = mv⃗  (1)

and hence

E = \frac{|p⃗|^{2}}{2m} .  (2)

The Schrödinger equation is the quantum analogue of the equation (2). To derive it, we need Planck’s law, which asserts that energy is proportional to the temporal-frequency f (cycles per second):

E = hf

or equivalently, setting ℏ := \frac{h}{2π} and letting ν := 2πf be the temporalfrequency in radians per second:

E = ℏν.

Both h and ℏ are referred to as Planck’s constant.

We also need a variant of Planck’s law, namely de Broglie’s law that asserts that momentum is proportional to the spatial-frequency k⃗ (cycles per unit length):

p⃗ = hk⃗

or equivalently, setting ξ⃗:= 2πk⃗ to be the spatial-frequency (in radians per unit length)

p⃗ = ℏξ.

Inserting these into (2) we obtain

ν = \frac{ℏ}{2m}|ξ⃗|^{2}.

We apply this to our wave function ϕ:

νϕ = \frac{ℏ}{2m}|ξ⃗|^{2}ϕ.

Now from Fourier analysis we know that multiplication by frequency corresponds to differentiation:

∂_{t}ϕ = −iνϕ; ∇ϕ = −iξϕ

and hence we arrive at the free Schrödinger equation

i∂_{t}ϕ + \frac{ℏ}{2m}Δϕ = 0.

If one added a potential V (x), then the energy becomes

E = \frac{|p⃗|^{2}}{2m} + V (x)

and then we obtain the Schrödinger equation with potential

i∂_{t}ϕ + \frac{ℏ}{2m}Δϕ − \frac{1}{ℏ}V (x)ϕ = 0.

For bound states with a fixed energy E, the equation becomes instead the time-independent Schrödinger equation with potential

−\frac{ℏ^{2}}{2m}Δϕ + V (x)ϕ = Eϕ.

More generally, we have a heuristic dictionary between classical quantities and quantum ones:

E ↦ iℏ∂_{t}
p⃗ ↦ iℏ∇
v ↦ \frac{iℏ}{m}∇
x ↦ x
…

Indeed, every classical observable A should be associated with some operator Op(A), and the expected value of this observable when applied to a quantum wave function ψ should be ⟨Op(A)ψ, ψ⟩. Thus for instance the total momentum should be ∫ ℏ Im(∇ψ\overline{ψ}) dx.

Unfortunately there is a limit as to how far this dictionary can be extended due to the non-commutativity of iℏ∇ and x. This is why quantum mechanics does not quite match up with classical mechanics except in the classical limit ℏ → 0.

2. Connection with relativity

The Schrödinger equation is not a relativistic equation - the speed of light c does not appear anywhere in the derivation, and the equations (1) are certainly non-relativistic. However, the Planck and de Broglie laws are highly compatible with relativity; indeed, assuming special relativity, one of these laws will imply the other. In the case of massless radiation m = 0, this can be seen from the formulae ν = |ξ|c (which reflects the fact that such radiation moves at c) and Einstein’s relation E = |p⃗|c.

More generally, for massive particles, the relativistic analogue of the Schrödinger equation is the Klein-Gordon equation, obtained by starting with Einstein’s energy-momentum relation

E^{2} = m^{2}c^{4} + |p⃗|^{2}c^{2}

(the Lorentz-invariant extension of Einstein’s famous equation E = mc^{2}) to obtain

−∂_{tt}ψ + c^{2}Δψ = \frac{m^{2}c^{4}}{ℏ^{2}} ψ.

Department of Mathematics, UCLA, Los Angeles, CA 90024

  E-mail address: tao@@math.ucla.edu