The 1D simple harmonic oscillator (in suitable units) is described by the Hamiltonian Ĥ = p²/2+x²/2. A particular state of the harmonic oscillator, called |λ), is defined as |A) = Ñ exp(Aâ³)|0), where Ñ = e¯||² is an overall normalisation constant, [0) denotes the ground state and â and â are the usual lowering/raising operators with [â, â] = 1. (a) By series expanding exp(Xât) prove that |A) is an eigenstate of â with eigenvalue X. (b) Using the result in part (a), show the 'overlap between two generally different states |X') and X is: \\'\\) = e− (\\'²+|A|² —2Ã'A)
The 1D simple harmonic oscillator (in suitable units) is described by the Hamiltonian Ĥ = p²/2+x²/2. A particular state of the harmonic oscillator, called |λ), is defined as |A) = Ñ exp(Aâ³)|0), where Ñ = e¯||² is an overall normalisation constant, [0) denotes the ground state and â and â are the usual lowering/raising operators with [â, â] = 1. (a) By series expanding exp(Xât) prove that |A) is an eigenstate of â with eigenvalue X. (b) Using the result in part (a), show the 'overlap between two generally different states |X') and X is: \\'\\) = e− (\\'²+|A|² —2Ã'A)
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![The 1D simple harmonic oscillator (in suitable units) is described by the Hamiltonian
Ĥ = p²/2+x²/2. A particular state of the harmonic oscillator, called |λ), is defined as
|A) = Ñ exp(Aâ³)|0),
where Ñ = e¯||² is an overall normalisation constant, [0) denotes the ground state and
â and â are the usual lowering/raising operators with [â, â] = 1.
(a) By series expanding exp(Xât) prove that |A) is an eigenstate of â with eigenvalue X.
(b) Using the result in part (a), show the 'overlap between two generally different states
|X') and X is:
\\'\\) = e− (\\'²+|A|² —2Ã'A)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2437b9d-e2ad-49ef-bf1e-5f1a1e22adaa%2F7d1efde0-037b-48db-9acf-0ce91eb0a803%2F61r22l8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The 1D simple harmonic oscillator (in suitable units) is described by the Hamiltonian
Ĥ = p²/2+x²/2. A particular state of the harmonic oscillator, called |λ), is defined as
|A) = Ñ exp(Aâ³)|0),
where Ñ = e¯||² is an overall normalisation constant, [0) denotes the ground state and
â and â are the usual lowering/raising operators with [â, â] = 1.
(a) By series expanding exp(Xât) prove that |A) is an eigenstate of â with eigenvalue X.
(b) Using the result in part (a), show the 'overlap between two generally different states
|X') and X is:
\\'\\) = e− (\\'²+|A|² —2Ã'A)
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