a) Show that the commutator [a", a*] = aat - a*a= 1 b) Show that the commutator [H, a*] = ħoa* (c) Using the identity obtained in (b), show that a"yn) is an eigen state of H with eigen energy ho( n+1+ 1 a + 2 a*a |Vn) = n[Wn) Helpful equations: H = ħo %3D
a) Show that the commutator [a", a*] = aat - a*a= 1 b) Show that the commutator [H, a*] = ħoa* (c) Using the identity obtained in (b), show that a"yn) is an eigen state of H with eigen energy ho( n+1+ 1 a + 2 a*a |Vn) = n[Wn) Helpful equations: H = ħo %3D
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![The raising (a") and lowing (a) operators associated with a simple harmonic oscillator
Hamiltonian (H) are given by
ax-ißp
ax+ißp
a
a
where a and B are real constants with aß
1
and [x.p] = i .
(a) Show that the commutator a", a*] = a a* - a*a = 1
(b) Show that the commutator [H,a*] = ħoa+
(c) Using the identity obtained in (b), show that a"yn) is an eigen state of H with eigen
energy ho n+1+-
2
1
Helpful equations: H = ħo a*a
a*a |Vn) = n\Wn}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4586fa80-8acb-4dfa-93c9-2eada7023384%2F478b9baf-13e7-46b5-adb0-7fc455b4b240%2Fosbf4ze_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The raising (a") and lowing (a) operators associated with a simple harmonic oscillator
Hamiltonian (H) are given by
ax-ißp
ax+ißp
a
a
where a and B are real constants with aß
1
and [x.p] = i .
(a) Show that the commutator a", a*] = a a* - a*a = 1
(b) Show that the commutator [H,a*] = ħoa+
(c) Using the identity obtained in (b), show that a"yn) is an eigen state of H with eigen
energy ho n+1+-
2
1
Helpful equations: H = ħo a*a
a*a |Vn) = n\Wn}
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