[ANSWERED] Consider three angular momentum operators Ji, I 1, 2, 3 with corresponding eigenstates I;, m, J…

Consider three angular momentum operators Ji, I 1, 2, 3 with corresponding eigenstates I;, m, J, as well as the total anguar momentum I *i 3- *2+*3 with corresponding states
, rri
J
).
Given f1 *. *2 3/2, and f3 3, what is the minimal value for
?
We start with a state in position representation (z). The translation operator
Tq
e
‘° ‘ translates the state /(z) bye n, ice.
Tz px)
/(z — n), and
P
denotes the parity operator. Evaluate the expression
Tb!° Tz
/(z).
Given a Hamiltonian
H ——
p
2
/2m + V(z) with initially unspecified U(z), and an initially unspecified operator O with
)H, O
0. State an operator O which guarantees that the Bloch theorem applies to
H.
Consider the excited state of a two-electron atom in which one of the electrons is in state u 1s* 0, rn 0, and the other electron in state n 2, 1 1, m 0. State the formal expression for the expectation value for the Coulomb energy between the two electrons in
the singled spin state.
How many permutations of a system of three fermions exist (including the identity operation)? How many of these permutations come with the eigenvalue —1?
In a system in which the Hamiltonian commutes with the time-reversal operator, can the wave function of an eigenstate to a non-degenerate eigenvalue be of the form /(z) Q(z) + ip(z), in which Q and y are linearly independent real and non-zero functions of z?
Consider the spin operators Si and S2 Of two spin-1/2 particles with angular momentum eigen- states si 1/2, mq, and s2 = 1/2, mq J, respectively, as well as the total angular momentum operator I = Si +S2 with angular momentum eigenstates
), ›).
Given s
1
1/2 and s
2
1/2, write down all possible combinations of and
rri .
The state
j ——
1, my 1) can immediately be expressed in terms of the •+ = 1/2, mq,) | 2 = 1/2, mq state vectors: li 1, my 1) |u = 1/2, mq, 1/2J s
2
1/2, mq = 1/2). Write down a corresponding expression for the state 1,
rri
y
—1).
Construct the state 1,
rri
y
0J, expressed in terms of the s
1
1/2, sql s
2
1/2, mq state vectors, by application of
Jr
S +
S
z—
to
j
——
1, my 1).
Him:
Apply Jr Io ltte
J
,
rri
)
bnsis
rind
1
+
S2
Io ltte
s
,
sql s2,
qq)
basis.
Use that your result for
——
1, my 0) obtained in (c) and 0,
rn
a
0J are orthonor- mal, and contain the same s
1
, mq, s
2
, mq ) states, to express 0,
rri
0J in terms of the si = 1/2, rug, 52 = 1/2, mq ) states.
Hinl:
The
ladder
operalor
Ja
acls
on Ihe
stnte
p
,
rri
)
in
Ihe
following
way:q
Ja
j, ml)
Ihe operatora S
and
S2—.
Consider a particle that moves in a double well potential, described by the Hamiltonian
+ vo,
a, b > o.
Show that >o is invariant imder the purity operation
Õ rim
well as under the time reversal operation ‘F.
We add a perturbation
to
Up, such that if – >o + *U. Here c > 0, and à is a small real and dimensionless parameter (note that the term òU renders the Hamiltonian Ú non-Hermitian, such that oomplex eigenvalues may occur). Determine whether if is invariant under the
Õ
and ‘ operations individually, as well as under the combined operation RÉ.
Suppose that {if, = 0, and that |Çy) is an eigenstate of žf with eigenvalue
E.
Show that, uiider these oonditions, also |Çp) is as eigenstate of if, and determine the corre- sponding eigenvalue Ê. How are
E
and Ê related in case that
|up) and |uy) describe the same quantum state?
|up) and |uy) are linearly independent, i.e. describe different quantum statee?
Suppose that the spectrum of the unperturbed Hamiltonian Up is non-degenerate and that the perturbation òÛ is turned on slowly by increasing z from zero, such that the eigenvalues E(à) are contínous functions of A. What follows for the spectrum, given that z remains small enough such that the eigenvalues E(A) remain non-degenerate?
Not
suppose that we enoounter complex eigenvalues E(A) for A > . What follows for the spectrum at A = ?
Determine the eigenvalues of the Hamiltonian
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