Q7:4 In the Lecture notes we showed that the 'annihilation’ operator for the harmonic oscillator is mw 1/2 ip a = D_ 2h V2mwh where r is the 'position operator' andp the momentum operator, and the creation operator is 1 a* = D. - () ip /2mwħ 2h (a) Show that a' is the adjoint operator of a. (b) Show that ħwa'a = H – 2 where H = +}mw?x² is the Hamiltonian operator for the oscillator. Note [r, p] = ih. (c) Show that a, a' = (d) From the above H = hw (a'a + ;) , hence show that [H, a] = –ħwa and H, a = ħwa*. You may assume [AB,C] = A[B,C] + [A, C]B.
Q7:4 In the Lecture notes we showed that the 'annihilation’ operator for the harmonic oscillator is mw 1/2 ip a = D_ 2h V2mwh where r is the 'position operator' andp the momentum operator, and the creation operator is 1 a* = D. - () ip /2mwħ 2h (a) Show that a' is the adjoint operator of a. (b) Show that ħwa'a = H – 2 where H = +}mw?x² is the Hamiltonian operator for the oscillator. Note [r, p] = ih. (c) Show that a, a' = (d) From the above H = hw (a'a + ;) , hence show that [H, a] = –ħwa and H, a = ħwa*. You may assume [AB,C] = A[B,C] + [A, C]B.
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Quantum Theory
![Q7:4 In the Lecture notes we showed that the 'annihilation’ operator for the harmonic
oscillator is
mw
1/2
ip
a =
D_
2h
V2mwh
where r is the 'position operator' andp the momentum operator, and the creation operator
is
1
a* =
D. - ()
ip
/2mwħ
2h
(a) Show that a' is the adjoint operator of a.
(b) Show that
ħwa'a = H –
2
where H = +}mw?x² is the Hamiltonian operator for the oscillator.
Note [r, p] = ih.
(c) Show that a, a' =
(d) From the above
H = hw (a'a + ;) ,
hence show that [H, a] = –ħwa and H, a = ħwa*.
You may assume [AB,C] = A[B,C] + [A, C]B.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff258f736-2efe-4abc-a546-b1bd73d984a7%2F7f0db8be-df12-4bb1-a95e-ed022110a89b%2F8txfd5m_processed.png&w=3840&q=75)
Transcribed Image Text:Q7:4 In the Lecture notes we showed that the 'annihilation’ operator for the harmonic
oscillator is
mw
1/2
ip
a =
D_
2h
V2mwh
where r is the 'position operator' andp the momentum operator, and the creation operator
is
1
a* =
D. - ()
ip
/2mwħ
2h
(a) Show that a' is the adjoint operator of a.
(b) Show that
ħwa'a = H –
2
where H = +}mw?x² is the Hamiltonian operator for the oscillator.
Note [r, p] = ih.
(c) Show that a, a' =
(d) From the above
H = hw (a'a + ;) ,
hence show that [H, a] = –ħwa and H, a = ħwa*.
You may assume [AB,C] = A[B,C] + [A, C]B.
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