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unscientific
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Homework Statement
Part (a): What is momentum operator classically and in quantum?
Part (b): Show the particle has 0 angular momentum.
Part (c): Determine whether angular momentum is present along: (i)z-axis, (ii) x-axis and find expectation values <Lz> and <Lx>.
Part (d): Find the result of finding Lz then Lx.
Part (e): Find the result of finding Lx then Lz.
Homework Equations
The Attempt at a Solution
Part (a)
Classically, ##\vec L = \vec r x \vec p##.
Quantum mechanically, ##\vec L = -i\hbar \vec r x \vec \nabla##.
Part (b)
Since ##L^2## is the angular part of ##\nabla^2 = -\frac{1}{sin \theta}\frac{\partial}{\partial \theta}\left(sin \theta \frac{\partial}{\partial \theta}\right) + \frac{1}{sin^2\theta}\frac{\partial^2}{\partial \theta^2}##, it has no radial dependence.
So ##\langle \psi|L^2|\psi\rangle = 0##.
Part (c)
Along z-axis, it is zero, since ##L_z = -i\hbar \frac{\partial}{\partial \phi}##.
Along x-axis, it is non-zero, since ##L_x## is a function of both ##(\theta, \phi)##.
Expectation value ##\langle L_z\rangle = 0##.
I'm not sure how to find the expectation value ##\langle L_x \rangle## without using brute force integration. Is there a trick somewhere I've missed?The rest of the question boggles me very much. Would appreciate any help!