2. Consider the raising and lowering operators of a one-dimensional harmonic oscil- lator, â = √√(q + p) and ↠= √(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  = 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 ↠= √(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  = â¹â is the number operator. (c) Show that |a) := e-la²/2ea¹ |0) is an eigenstate of â with eigenvalue a.
Related questions
Question
![2. Consider the raising and lowering operators of a one-dimensional harmonic oscil-
m@P) and â†
√(-p), which satisfy
2ħ
mω
lator, 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F466dfe05-514c-4e8d-9e28-eee295fc0d76%2F561f4d5e-5f3f-496a-9c49-97ac4e31f9eb%2F0v0ijnl_processed.png&w=3840&q=75)
Transcribed Image Text:2. Consider the raising and lowering operators of a one-dimensional harmonic oscil-
m@P) and â†
√(-p), which satisfy
2ħ
mω
lator, 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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 1 images
