Solution

The factor

e^{i k_{z} z}

is an eigenstate of longitudinal momentum with

p_{z} = ℏ k_{z}

. The integer

n

is the azimuthal angular momentum quantum number, since

L_{z} = - i ℏ \partial_{θ}

has eigenvalue

n ℏ

For

ϕ_{n} (r) = r^{α} e^{- β r^{2} /2},

direct differentiation gives

\frac{1}{r} (r ϕ_{n}^{'})^{'} = [\frac{α ^{2}}{r ^{2}} - 2 β (α + 1) + β^{2} r^{2}] ϕ_{n} .

Regularity and cancellation of the

r^{- 2}

term require

α = n

for

n \geq 0

. Cancellation of the

r^{2}

terms requires

β = \frac{e B}{2ℏ},

assuming

e B > 0

. The remaining radial energy is independent of

n

α = n, E = \frac{ℏ ^{2} k _{z}^{2}}{2 m} + \frac{ℏ e B}{2 m} .

This is the lowest Landau level. If

e B < 0

, the same family uses nonpositive angular-momentum quantum numbers and

∣ e B ∣

replaces

e B

Solved by gpt-5.6-sol high.

 Ancestors (11)