(a.) Some energy levels are degenerate. For example, E = 2ħw can be achieve with (nx, ny) = (1, 0); (0, 1). This energy level has a degeneracy D(2ħw) = 2. What is the degeneracy of energy level E = Nhw (where N is a positive integer)? (12,0) + 2 |1, 1) + 10, 2)) (c.) Calculate (Ĥ), (px), (py), and (âŷ) for the state above. (b.) Consider the state (0)) = √6 What is (t)) at a later time t > 0?
(a.) Some energy levels are degenerate. For example, E = 2ħw can be achieve with (nx, ny) = (1, 0); (0, 1). This energy level has a degeneracy D(2ħw) = 2. What is the degeneracy of energy level E = Nhw (where N is a positive integer)? (12,0) + 2 |1, 1) + 10, 2)) (c.) Calculate (Ĥ), (px), (py), and (âŷ) for the state above. (b.) Consider the state (0)) = √6 What is (t)) at a later time t > 0?
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Transcribed Image Text:Consider the 2D harmonic oscillator Hamiltonian:
87 - 12/24 (13² + 8² ) + 2 ²³² (1² + 8² )
Ĥ
mw²
2m
Unless otherwise specified, we will work in the eigenstates that satisfy:
Ĥ|nz, ny) = Enz,ny |nx, ny)
x,
with Eng,ny = ħw(nx + Ny + 1).
(a.) Some energy levels are degenerate. For example, E 2ħw can be achieve with (nx, ny) = (1, 0); (0, 1).
This energy level has a degeneracy D(2ħw) = 2. What is the degeneracy of energy level E
(where N is a positive integer)?
=
Nhw
(b.) Consider the state (0)) = √ (12,0) + 2 |1, 1) + (0,2)).
(c.) Calculate (Ĥ), (px), (py), and (âŷ) for the state above.
=
What is (t)) at a later time t > 0?
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This problem is related to a two-dimensional harmonic oscillator that has degenerate energy levels. The angular frequency for both the dimension is the same which is equal to w.
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