Consider the ladder operators of the one-dimensional harmonic oscillator 1 mw -î + i- V 2h a = 2mwh mw 1 a+ 2h 2mwh' (a) Express the Hamiltonian H = p² /2m + mw?î²/2 in terms of operators a and a+ (b) Use the Hamiltonian obtained in (a) above to find the time evolution of a and a4 using d. [Ĥ , A] dt
Consider the ladder operators of the one-dimensional harmonic oscillator 1 mw -î + i- V 2h a = 2mwh mw 1 a+ 2h 2mwh' (a) Express the Hamiltonian H = p² /2m + mw?î²/2 in terms of operators a and a+ (b) Use the Hamiltonian obtained in (a) above to find the time evolution of a and a4 using d. [Ĥ , A] dt
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![**One-Dimensional Harmonic Oscillator: Ladder Operators**
Consider the ladder operators of the one-dimensional harmonic oscillator:
\[ a = \sqrt{\frac{m\omega}{2\hbar}} \, \hat{x} + i \frac{1}{\sqrt{2m\omega\hbar}} \, \hat{p} \]
\[ a_+ = \sqrt{\frac{m\omega}{2\hbar}} \, \hat{x} - i \frac{1}{\sqrt{2m\omega\hbar}} \, \hat{p} \]
**Tasks:**
(a) **Express the Hamiltonian \(\hat{H}\):**
The Hamiltonian is given by:
\[ \hat{H} = \frac{\hat{p}^2}{2m} + \frac{m\omega^2\hat{x}^2}{2} \]
Express this Hamiltonian in terms of the operators \(a\) and \(a_+\).
(b) **Time Evolution:**
Use the Hamiltonian obtained in part (a) to find the time evolution of \(a\) and \(a_+\) using the equation:
\[ \frac{d}{dt} \hat{A} = \frac{i}{\hbar} [\hat{H}, \hat{A}] \]
This involves calculating the commutator \([\hat{H}, \hat{A}]\) where \(\hat{A}\) can be \(a\) or \(a_+\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2F1a9a064a-5570-4fc8-9e8c-15e3e9b7ff26%2Fmmww67o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**One-Dimensional Harmonic Oscillator: Ladder Operators**
Consider the ladder operators of the one-dimensional harmonic oscillator:
\[ a = \sqrt{\frac{m\omega}{2\hbar}} \, \hat{x} + i \frac{1}{\sqrt{2m\omega\hbar}} \, \hat{p} \]
\[ a_+ = \sqrt{\frac{m\omega}{2\hbar}} \, \hat{x} - i \frac{1}{\sqrt{2m\omega\hbar}} \, \hat{p} \]
**Tasks:**
(a) **Express the Hamiltonian \(\hat{H}\):**
The Hamiltonian is given by:
\[ \hat{H} = \frac{\hat{p}^2}{2m} + \frac{m\omega^2\hat{x}^2}{2} \]
Express this Hamiltonian in terms of the operators \(a\) and \(a_+\).
(b) **Time Evolution:**
Use the Hamiltonian obtained in part (a) to find the time evolution of \(a\) and \(a_+\) using the equation:
\[ \frac{d}{dt} \hat{A} = \frac{i}{\hbar} [\hat{H}, \hat{A}] \]
This involves calculating the commutator \([\hat{H}, \hat{A}]\) where \(\hat{A}\) can be \(a\) or \(a_+\).
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