The Hamiltonian for the one dimensional quantum oscillator is1p²1Ĥ = 1² + ½ k²² = 12 + √ mw² à²2m2m2where k = mw².1) Define the operators ₁₁ and ₁₁ such that Ĥ = ½ħw (p² + ²). Define Ĥ2 asa function of 1 and p₁ such that Ĥ = hwĤ₂.-2) Let us define the new operators â (1 + i₁) and â† = 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 havethe same eigenstates, what are their corresponding eigenvalues?

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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 â† = 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 â† = ½(î₁ — iŷ1).