The Hamiltonian for the one dimensional quantum oscillator is 1 Н p² 1 + 2m 2 +mw²x² 2m where k = mw² 1) Define the operators ₁₁ and ₁₁ such that Ĥ = ½ħw(p² + î²). Define Ĥ₂ a a function of 1 and ₁ such that Ĥ = ħwĤ₂. - 2) Let us define the new operators â = 1/2(1 + iĝ₁) and ↠= ½(î₁ — iŷ1).
The Hamiltonian for the one dimensional quantum oscillator is 1 Н p² 1 + 2m 2 +mw²x² 2m where k = mw² 1) Define the operators ₁₁ and ₁₁ such that Ĥ = ½ħw(p² + î²). Define Ĥ₂ a a function of 1 and ₁ such that Ĥ = ħwĤ₂. - 2) Let us define the new operators â = 1/2(1 + iĝ₁) and ↠= ½(î₁ — iŷ1).
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![The Hamiltonian for the one dimensional quantum oscillator is
1
p²
1
Ĥ = 1² + ½ k²² = 12 + √ mw² à²
2m
2m
2
where k = mw².
1) Define the operators ₁₁ and ₁₁ such that Ĥ = ½ħw (p² + ²). Define Ĥ2 as
a function of 1 and p₁ such that Ĥ = hwĤ₂.
-
2) Let us define the new operators â (1 + i₁) and ↠= ½(î₁ — ip₁).
Express ₁ and p₁ as a function of â and â³. Knowing that [^^1,î₁] = i and
[1, 1] = -i, calculate âât and â†â. Express Ĥ2 as a function of a and at.
3) Let us define Ñ such that Ĥ₂ = Ñ + ½. Knowing that Ĥ, Ĥ₂ and Ñ have
the same eigenstates, what are their corresponding eigenvalues?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d6b1f03-b4bd-4f60-ae23-d8256607755b%2Fdea78957-124a-493b-b240-0b87878f2cd2%2Ffuf39p3_processed.png&w=3840&q=75)
Transcribed Image Text:The Hamiltonian for the one dimensional quantum oscillator is
1
p²
1
Ĥ = 1² + ½ k²² = 12 + √ mw² à²
2m
2m
2
where k = mw².
1) Define the operators ₁₁ and ₁₁ such that Ĥ = ½ħw (p² + ²). Define Ĥ2 as
a function of 1 and p₁ such that Ĥ = hwĤ₂.
-
2) Let us define the new operators â (1 + i₁) and ↠= ½(î₁ — ip₁).
Express ₁ and p₁ as a function of â and â³. Knowing that [^^1,î₁] = i and
[1, 1] = -i, calculate âât and â†â. Express Ĥ2 as a function of a and at.
3) Let us define Ñ such that Ĥ₂ = Ñ + ½. Knowing that Ĥ, Ĥ₂ and Ñ have
the same eigenstates, what are their corresponding eigenvalues?
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