The wavefunction of a particle is subjected to a sperically symmetric potential V(r) is given byb(x)= (xy 32)f(r),(1)where f is some function of the radius r alone.a) Is an eigenfunction of L2? If so, what is its l-value? If not, what are the possible values of / wemay obtain when L2 is measured.b) What are the probabilities for the particle to be found in various m states?c) Suppose it is known that b is an energy eigenfunction with energy eigenvalue E. Indicate how wemay find V(r).

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Asked Dec 3, 2019
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It's a quantum mechanics question.

The wavefunction of a particle is subjected to a sperically symmetric potential V(r) is given by
b(x)= (xy 32)f(r),
(1)
where f is some function of the radius r alone.
a) Is an eigenfunction of L2? If so, what is its l-value? If not, what are the possible values of / we
may obtain when L2 is measured.
b) What are the probabilities for the particle to be found in various m states?
c) Suppose it is known that b is an energy eigenfunction with energy eigenvalue E. Indicate how we
may find V(r).
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The wavefunction of a particle is subjected to a sperically symmetric potential V(r) is given by b(x)= (xy 32)f(r), (1) where f is some function of the radius r alone. a) Is an eigenfunction of L2? If so, what is its l-value? If not, what are the possible values of / we may obtain when L2 is measured. b) What are the probabilities for the particle to be found in various m states? c) Suppose it is known that b is an energy eigenfunction with energy eigenvalue E. Indicate how we may find V(r).

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Expert Answer

Step 1

The relation between the Cartesian and spherical coordinates is given as,

x = rsin Ocos O
y =r sinO sino
2 =r cos0
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x = rsin Ocos O y =r sinO sino 2 =r cos0

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Step 2

Here θ is the polar angle and φ is the azimuthal angle.

 

Use the spherical polar coordinates to rewrite the wave function ψ in terms of r, θ, φ as,

 

w(r,0,0)= (r sin@ coso+rsin Osin o+3rcos 0) f (r)
= (sin0cos o + sin@sin o+ 3 cos 0)rf (r)
= Y (0.9) R, (r)
Ym (0,0) = (sin0cos o + sin O sin o+3cos 0)
R, (r) =rf (r)
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w(r,0,0)= (r sin@ coso+rsin Osin o+3rcos 0) f (r) = (sin0cos o + sin@sin o+ 3 cos 0)rf (r) = Y (0.9) R, (r) Ym (0,0) = (sin0cos o + sin O sin o+3cos 0) R, (r) =rf (r)

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Step 3

Here Ylm represents spherical harmonics and Rnl (r) represents the radial part of the wave function.

 

a)

The operator...

1ô sin@.
L' = -h?
( sin@ col
0 0
sin? 0 ôg²
дө
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1ô sin@. L' = -h? ( sin@ col 0 0 sin? 0 ôg² дө

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