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2. Work out [â, 6]. Simplify to the maximum extent possible. 3. Work out âb. 4. Work out ba

2. Work out [â, 6]. Simplify to the maximum extent possible. 3. Work out âb. 4. Work out ba

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2,3,4, and 5 please

In classical mechanics it is common to define variables that are linear combinations of position variables and momentum variables. The same useful combinations found in classical mechanics turn up useful in quantum mechanics too! Consider a 1D problem. Given the operators £ and p, such that [2, p] = ihî. Define two new operators: â = câ + idp and b = c£ - idp where c and d are real constants. Answer the following: Prove that the adjoint of â is equal to b. Work out [â, 6]. Simplify to the maximum extent possible. 2. 3. Work out âb. Work out bâ 5. Add âb with bâ and compare this result to the Hamiltonian for the quantum simple harmonic oscillator (QSHO) that is given as: A = +! maximumly match the Hamiltonian for the QSHO to the combination of (âb + bâ). Explain in words what parts match and what parts do not match. Propose what the values for c and d should be in order to 2m 2
Transcribed Image Text:In classical mechanics it is common to define variables that are linear combinations of position variables and momentum variables. The same useful combinations found in classical mechanics turn up useful in quantum mechanics too! Consider a 1D problem. Given the operators £ and p, such that [2, p] = ihî. Define two new operators: â = câ + idp and b = c£ - idp where c and d are real constants. Answer the following: Prove that the adjoint of â is equal to b. Work out [â, 6]. Simplify to the maximum extent possible. 2. 3. Work out âb. Work out bâ 5. Add âb with bâ and compare this result to the Hamiltonian for the quantum simple harmonic oscillator (QSHO) that is given as: A = +! maximumly match the Hamiltonian for the QSHO to the combination of (âb + bâ). Explain in words what parts match and what parts do not match. Propose what the values for c and d should be in order to 2m 2
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