2. Consider the raising and lowering operators of a one-dimensional harmonic oscil-m@P) and â†√(-p), which satisfy2ħmωlator, â = √√(q +2ħ=â|n) = √n|n − 1),at |n) = √n + 1/n + 1).(a) Show that [Ñ, â] = −â and [Ñ‚ â†] = â†, where Ñ = â¹â is the number operator.(c) Show that |a) := e¯\α|²/²µ¤â¹ |0) is an eigenstate of â with eigenvalue a.

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2. Consider the raising and lowering operators of a one-dimensional harmonic oscil- lator, â = √√(q + p) and â† = √(q − p), which satisfy Μω 2ħ mw. mw. â|n) = √nn - 1), â¹ |n) = √n + 1|n + 1). (a) Show that [Ñ, â] = −â and [Ñ‚ â†] = â†, where Â = â¹â is the number operator. (c) Show that |a) := e-la²/2ea¹ |0) is an eigenstate of â with eigenvalue a.

2. Consider the raising and lowering operators of a one-dimensional harmonic oscil- lator, â = √√(q + p) and â† = √(q − p), which satisfy Μω 2ħ mw. mw. â|n) = √nn - 1), â¹ |n) = √n + 1|n + 1). (a) Show that [Ñ, â] = −â and [Ñ‚ â†] = â†, where Â = â¹â is the number operator. (c) Show that |a) := e-la²/2ea¹ |0) is an eigenstate of â with eigenvalue a.