2. For a certain system the operator corresponding to the physical quantity A does not commute with the Hamiltonian. It has eigenvalues a₁ and a2, corresponding to eigenfunctions: Φι = (u1 + u2) √2 " Φ2 = (u1 - u2) √2 where u1 and U2 are eigenfunctions of the Hamiltonian, with eigenvalues E1 and E2. If the system is in the state =₁ at time t=0, show that the expectation value of A at time t is: (A) = = a1 + a2 + 2 a1 a2 (E1 – E2)t 2 COS h Hints: Remember that the time dependent SE has solutions which include an exponential phase factor exp(-iwt). Start with: (A) = |c₁|²a1 + |c2|2a2 Find the cs by considering the time evolution of the wave function to show that |ci|² == | | 1 [1 + cos(w1 − w2)t] = 2 (eix + e −ix) COST = 2
2. For a certain system the operator corresponding to the physical quantity A does not commute with the Hamiltonian. It has eigenvalues a₁ and a2, corresponding to eigenfunctions: Φι = (u1 + u2) √2 " Φ2 = (u1 - u2) √2 where u1 and U2 are eigenfunctions of the Hamiltonian, with eigenvalues E1 and E2. If the system is in the state =₁ at time t=0, show that the expectation value of A at time t is: (A) = = a1 + a2 + 2 a1 a2 (E1 – E2)t 2 COS h Hints: Remember that the time dependent SE has solutions which include an exponential phase factor exp(-iwt). Start with: (A) = |c₁|²a1 + |c2|2a2 Find the cs by considering the time evolution of the wave function to show that |ci|² == | | 1 [1 + cos(w1 − w2)t] = 2 (eix + e −ix) COST = 2
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![2. For a certain system the operator corresponding to the physical quantity A does not
commute with the Hamiltonian. It has eigenvalues a₁ and a2, corresponding to
eigenfunctions:
Φι
=
(u1 + u2)
√2
"
Φ2
=
(u1 - u2)
√2
where u1 and U2 are eigenfunctions of the Hamiltonian, with eigenvalues E1 and E2.
If the system is in the state =₁ at time t=0, show that the expectation value of A at
time t is:
(A) =
=
a1 + a2
+
2
a1 a2 (E1 – E2)t
2
COS
h
Hints: Remember that the time dependent SE has solutions which include an
exponential phase factor exp(-iwt).
Start with:
(A) = |c₁|²a1 + |c2|2a2
Find the cs by considering the time evolution of the wave function to show that
|ci|² == | |
1
[1 + cos(w1 − w2)t]
=
2
(eix + e −ix)
COST =
2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6713d75e-2fb3-4074-927a-cea8cea15561%2Ffe8a3b70-ec6f-4b3b-b1bb-f865ba2990f7%2Fo2m2z5i_processed.png&w=3840&q=75)
Transcribed Image Text:2. For a certain system the operator corresponding to the physical quantity A does not
commute with the Hamiltonian. It has eigenvalues a₁ and a2, corresponding to
eigenfunctions:
Φι
=
(u1 + u2)
√2
"
Φ2
=
(u1 - u2)
√2
where u1 and U2 are eigenfunctions of the Hamiltonian, with eigenvalues E1 and E2.
If the system is in the state =₁ at time t=0, show that the expectation value of A at
time t is:
(A) =
=
a1 + a2
+
2
a1 a2 (E1 – E2)t
2
COS
h
Hints: Remember that the time dependent SE has solutions which include an
exponential phase factor exp(-iwt).
Start with:
(A) = |c₁|²a1 + |c2|2a2
Find the cs by considering the time evolution of the wave function to show that
|ci|² == | |
1
[1 + cos(w1 − w2)t]
=
2
(eix + e −ix)
COST =
2
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