Separating variables for particle in XY plane with r-dependent potential

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gfd43tg
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Homework Statement


Printed homework question 2: a particle of mass μ moving in the xy plane under a central potential V(r), with a polar-coordinate sketch showing r and θ, the Hamiltonian H, and parts a) to d) asking about separation of variables, m, and angular momentum.
Printed homework problem parts e and f on a two-dimensional infinite circular well: piecewise potential V(r) equal to 0 for r less than a and infinity for r greater than a, the radial equation (21), eigenvalue formula E_nm, and a table of zeros x_nm for m = 0 to 3.

Text problem part g: "We use the notation s, p, d, f, ... to denote states with angular momentum quantum number m = 0, 1, 2, 3, ... respectively. What are the five lowest energy states in this notation (for example, a state with n = 0 and m = 0 should be denoted as 0s)? List them in increasing order of energy."

Typed homework problem parts h to k on a two-particle circular well, electron spin and repulsion, and a particle in a 3D infinite cylindrical well with energies and normalizable wavefunctions asked for.


Homework Equations

The Attempt at a Solution


a) I know that I am allowed to separate variables, such that ##\psi (r, \theta) = R(r) \Theta(\theta)##, but the part with the exponential is yet to be seen. So I put it into the Hamiltonian,
$$H = - \frac {\hbar^{2}}{2 \mu} \Big ( \frac {1}{R} \frac {\partial^{2} R}{\partial r^{2}} + \frac {1}{R} \frac {1}{r} \frac {\partial R}{\partial r} + \frac {1}{r^{2}} \frac {1}{\Theta} \frac {\partial^{2} \Theta}{\partial \theta^{2}} \Big ) + V $$

Since V is a function of r, it is not a constant, so I can't just call every term with derivatives as constants, can I? This is what I would normally be able to do, particularly if V = 0. So how do I proceed from here?
 
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You can take another angular derivative, then all the non-trivial R terms will go away.
The problem statement is misleading, there are solutions that cannot be written in that way, e.g. superpositions of states with different m.[/size]
 
What is the other angular derivative you are talking about? I just see the one in my equation.
 
Wouldn't that give some sort of triple derivative on the right side of my equation for the third term in the parenthesis?
 
It might help you to write out the complete equation.
$$-\frac{\hbar^2}{2\mu}\left(\frac{R''}{R} + \frac{R'}{rR} + \frac{\Theta'}{r^2\Theta}\right) + V = E.$$ Multiply by ##r^2## and you can separate the terms into those depending on ##r## and those that depend only on ##\theta##.
 
Okay, so I can separate the equation

$$ -(H-V)r^{2} \frac {2 \mu}{\hbar^{2}} - \frac {r^{2}}{R} R'' - \frac {r}{R} R' = \frac {\Theta''}{\Theta} $$
But how can I solve this?
 
The left side is independent of theta, which means the whole equation has to have the same value for all values of theta. You can set the left side equal to some constant (it will depend on r but not on theta). This is the same as setting the derivative (with respect to theta) to zero.