Calculate the three lowest energy levels, together with their degeneracies, for the following systems (assume equal-mass distinguishable particles). (a) Three noninteracting spin particles in a box of length L. (b) Four noninteracting spin particles in a box of length L.
Calculate the three lowest energy levels, together with their degeneracies, for the following systems (assume equal-mass distinguishable particles). (a) Three noninteracting spin particles in a box of length L. (b) Four noninteracting spin particles in a box of length L.
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![in the wire that produces the B held!
Problems
4.1 Calculate the three lowest energy levels, together with their degeneracies, for the
following systems (assume equal-mass distinguishable particles).
(a) Three noninteracting spin
(b) Four noninteracting spin
4.2 Let Ta denote the translation
particles in a box of length L.
particles in a box of length L.
operator (displacement vector d); let D(n,p) denote
the rotation operator (n and are the axis and angle of rotation, respectively); and
let denote the parity operator. Which, if any, of the following pairs commute?
Why?
(a) Ta and Ta (d and d' in different directions).
(b) D(n,p) and D(f', ') (f and fi' in different directions).
(c) Ta and .
(d) D(n,) and 7.
4.3 A quantum-mechanical state is known to be a simultaneous eigenstate of two
Hermitian operators A and B that anticommute:
AB+BA=0.
What can you say about the eigenvalues of A and B for state
point using the parity operator (which can be chosen to satisfy
the momentum operator.
4.4 A spin particle is bound to a fixed center by a spherically symmetrical potential.
(a) Write down the spin-angular function 1/2m=1/2
(b) Express (0-x) =1/2,m=1/2 in terms of some other y
(c) Show that your result in (b) is understandable in view of the transforma-
tion properties of the operator S.x under rotations and under space inversion
(parity).
? Illustrate your
== ¹) and
4.5 Because of weak (neutral-current) interactions, there is a parity-violating potential
between the atomic electron and the nucleus as follows:
V=A[83)(x)S.p+S.p8³)(x)],
Problems
where S and p are the spin and momentum operators of the electron, and the nu-
cleus is assumed to be situated at the origin. As a result, the ground state of an alkali
atom, usually characterized by In,1, j,m), actually contains very tiny contributions
from other eigenstates as follows:
\n,l, j,m) → \n,l,j,m) + Σ Crim'n'',j', m').
n'l' j'm'
v=f%20
V
301
On the basis of symmetry considerations alone, what can you say about (n',l', j',m'),
which give rise to nonvanishing contributions? Suppose the radial wave functions
and the energy levels are all known. Indicate how you may calculate Cn'' j'm'. Do
we get further restrictions on (n',l', j',m')?
4.6 Consider a symmetric rectangular double-well potential:
for x > a+b;
for a < x <a+b;
Vo> 0 for x <a.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F283a8bd9-9fd8-47a2-9f95-ef863f6a6090%2F0641655a-a676-4d9b-8f6e-ea2956feac0d%2Fwt0zwbr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:in the wire that produces the B held!
Problems
4.1 Calculate the three lowest energy levels, together with their degeneracies, for the
following systems (assume equal-mass distinguishable particles).
(a) Three noninteracting spin
(b) Four noninteracting spin
4.2 Let Ta denote the translation
particles in a box of length L.
particles in a box of length L.
operator (displacement vector d); let D(n,p) denote
the rotation operator (n and are the axis and angle of rotation, respectively); and
let denote the parity operator. Which, if any, of the following pairs commute?
Why?
(a) Ta and Ta (d and d' in different directions).
(b) D(n,p) and D(f', ') (f and fi' in different directions).
(c) Ta and .
(d) D(n,) and 7.
4.3 A quantum-mechanical state is known to be a simultaneous eigenstate of two
Hermitian operators A and B that anticommute:
AB+BA=0.
What can you say about the eigenvalues of A and B for state
point using the parity operator (which can be chosen to satisfy
the momentum operator.
4.4 A spin particle is bound to a fixed center by a spherically symmetrical potential.
(a) Write down the spin-angular function 1/2m=1/2
(b) Express (0-x) =1/2,m=1/2 in terms of some other y
(c) Show that your result in (b) is understandable in view of the transforma-
tion properties of the operator S.x under rotations and under space inversion
(parity).
? Illustrate your
== ¹) and
4.5 Because of weak (neutral-current) interactions, there is a parity-violating potential
between the atomic electron and the nucleus as follows:
V=A[83)(x)S.p+S.p8³)(x)],
Problems
where S and p are the spin and momentum operators of the electron, and the nu-
cleus is assumed to be situated at the origin. As a result, the ground state of an alkali
atom, usually characterized by In,1, j,m), actually contains very tiny contributions
from other eigenstates as follows:
\n,l, j,m) → \n,l,j,m) + Σ Crim'n'',j', m').
n'l' j'm'
v=f%20
V
301
On the basis of symmetry considerations alone, what can you say about (n',l', j',m'),
which give rise to nonvanishing contributions? Suppose the radial wave functions
and the energy levels are all known. Indicate how you may calculate Cn'' j'm'. Do
we get further restrictions on (n',l', j',m')?
4.6 Consider a symmetric rectangular double-well potential:
for x > a+b;
for a < x <a+b;
Vo> 0 for x <a.
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