Question

Write down an expression for the z-component of angular momentum, Lz, of a particle moving in the (x, y) plane in terms of its linear momentum components px and py .

Using the operator correspondence px = −iℏ(∂/∂x) etc., show that

Lz = −iℏ( x(∂/∂y) − y(∂/∂x))

Hence show that Lz = −iℏ(∂/∂ϕ), where the coordinates (x,y) and (r, ϕ) are related in the usual way.

Assuming that the wave function for this particle can be written in the form ψ(r, ϕ) = R(r)Φ(ϕ) show that the z-component of angular momentum is quantized with eigen value ℏ, where m is an integer.

Answer 1

From the definition of angular momentum

If θ is the polar angle, ϕ the azimuthal angle and r the radial distance, (Fig. 3.26). Then

Substituting (2), (6), (7) and (8) in (9) and simplifying, the first two terms drop *** and the third one reduces to ∂ψ/∂ϕ, yielding

In Problem 3.15 it was shown that the Schrodinger equation was separated into radial (r) and angular parts (θ and ϕ). The angular part was shown to be separated into θ and ϕ components. The solution to ϕ was shown to be

where m is an integer.

Thus the z-component of angular momentum is quantized with eigen value ℏ.