Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

Dimensional Analysis of the Schrödinger Equation

The Schrödinger equation for an electron in a hydrogen atom is

where is Planck's constant (from the uncertainty principle), m is the mass of the electron, and K is a combination of other constants (K/r is the potential energy).

Use these dimensioned parameters of the system to create a constant with the dimensions of energy.

Solution

Our job here is to construct a constant with the dimensions of energy from the constants that appear in the Schrödinger above: , m, and K. These have dimensions:

[] = [px] = (ML/T)(L) = ML2/T (using the uncertainty principle)

[m] = M

[K] = [Energy * r] = (ML2/T2)(L) = ML3/T2 (from the fact that K/r is an energy)

We can take a product of any power of these to make an energy:

[α mβ Kγ ] = Mα+β+γ L2α+3γ T-α - 2γ

To make this product into an energy, which has dimensions ML2/T2, we have therefore to choose:

α + β + γ = 1
2α + 3γ = 2
−α − 2γ = −2.

Solving these 3 equations in 3 unknowns is straightforward and yields the result:

α = −2
β = 1
γ = 2

This gives us our result for an energy scale, E0:

E0 = mK2/2.

For the case of the hydrogen atom, the Schröinger equation looks like this with K = kCe2, where kC = 1/4πε0, the Coulomb constant. This particular combination differs from the standard quantum unit of atomic energy, the Rydberg, R, by a factor of 2.

Notice that we have only been asked to find a constant with dimensions of energy. We have not been asked to make the Schröinger equation dimensionless (which would involve finding natural scales for the variables, r, t, and for the wave function ψ), and we have not been asked to solve the Schröinger equation.

[GRADING PATTERN: Correctly identifying the dimensions of and K were worth 2 pts each, as was getting right the dimensions of energy. The setup -- figuring out how the product of various constants would combine -- was worth 4 more. Working out a correct solution was worth 5.]

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