- #1
spacetimedude
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Homework Statement
The wave function of a particle is known to have the form $$u(r,\theta,\phi)=AR(r)f(\theta)\cos(2\phi)$$ where ##f## is an unknown function of ##\theta##. What can be predicted about the results of measuring
(a) the z-component of angular momentum;
(b) the square of the angular momentum?
Homework Equations
##cos(2\phi)={exp(2i\phi)+exp(-2i\phi)}/2##
Also, I know the spherical harmonics and the corresponding eigenvalues.
The solutions are:
a) ##L_z=+/- 2## with equal probability.
b) ##l\geq 2## but nothing can be said if we don't know ##f(\theta)##
where l denotes the angular momentum quantum number.
The Attempt at a Solution
For the first one, I am trying to make sense of what the physical interpretation of the two exponential functions is in the wave function. Why is ##L_z=+/- 2##? Is u a linear combination of two states, ##exp(2i\phi)## and ##exp(-2i\phi)## and do the value -2 and 2 indicate the angular momentum? If I act some operator to u, do we only consider one of the exponential state? I am a bit rusty with the meaning of a wave function.
The second one, I just know that $$\hat{L^2} |l,m>=\hbar^2 l(l+1)|l,m>$$.
Thanks in advance.