7. Find A, by normalizing |n). Suggestion: Compute (n|n) for n = 0, 1, 2,3 then find the pattern. Use the identity (Qflg) = (SIQ†|9) for Q = ât. 8. Show that |n) are the eigenvectors of H, i.e., Ĥ \n) = En \n) (7) is satisfied. Find the energy eigenvalue En. Knowing that Ĥ is an observable, what can you tell about (m|n) for m + n? 9. Using your results, show that the following formula holds: ât |n) = Vn+1 |n+1) â n) = Vn In – 1) (8a) (8b) Do this either by giving a general proof or by showing that they hold for n = = 0,1,2,3. 10. Let's go back to Eq.(2). What is the SI unit of the coefficient ? Does it make 2mw sense to you? 11. Show that the square of position operator is (âtât + â†â + â↠+ ââ) 2mw (9) 12. Compute (0|£|0) and (0|â²|0). This can be done very efficiently if you use Eq.(8) 13. Find the expression of the momentum operator square, p, in terms of the Fock oper- ators. Compute (Op|0) and (0|* |0). The Fock operator â is defined by (1) || 2h where î and p are the position and momentum operators, respectively. 1. Write dowm ât in terms of î and p. 2. Show that (ât +â) 2mw (2) p=i 2 ħmw (ât – â) (3) hold. 3. Show that the cannonical communation relation, [â, p = iħ, yields the so-called bosonic commutation relation, (â, ât] = 1. (4) 4. Show that the Hamiltonian of the SHO, is written as 2 2m (*+) Ĥ = hw Ñ + (5) where N = âtâ is called the number operator. 5. Show that Ñ is Hermitian. Suggestion: Use the identity from Exercise #1, (QR)† = 6. A normalized vector |0) (so that (0|0) = 1) is defined to satisfy â|0) = 0. With this the following vectors are constructed: |n) = A, (ât)" |0) for n = = 0, 1,2, ... (6) where An are constant with A, = 1. Compute Nn) for n = are the eigenvectors of N, i.e., Ñ|N) is proportional to |N). Find the eigenvalues of N from the proportionality. 0, 1,2,3 to show that these CS Scanned with CamScanner

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Please answer #12 and #14.  The first page is provided for context.

7. Find A, by normalizing |n). Suggestion: Compute (n|n) for n = 0, 1, 2,3 then find the
pattern. Use the identity (Qflg) = (SIQ†|9) for Q = ât.
8. Show that |n) are the eigenvectors of H, i.e.,
Ĥ \n) = En \n)
(7)
is satisfied. Find the energy eigenvalue En. Knowing that Ĥ is an observable, what
can you tell about (m|n) for m + n?
9. Using your results, show that the following formula holds:
ât |n) = Vn+1 |n+1)
â n) = Vn In – 1)
(8a)
(8b)
Do this either by giving a general proof or by showing that they hold for n =
= 0,1,2,3.
10. Let's go back to Eq.(2). What is the SI unit of the coefficient
? Does it make
2mw
sense to you?
11. Show that the square of position operator is
(âtât + â†â + â↠+ ââ)
2mw
(9)
12. Compute (0|£|0) and (0|â²|0). This can be done very efficiently if you use Eq.(8)
13. Find the expression of the momentum operator square, p, in terms of the Fock oper-
ators.
Compute (Op|0) and (0|* |0).
Transcribed Image Text:7. Find A, by normalizing |n). Suggestion: Compute (n|n) for n = 0, 1, 2,3 then find the pattern. Use the identity (Qflg) = (SIQ†|9) for Q = ât. 8. Show that |n) are the eigenvectors of H, i.e., Ĥ \n) = En \n) (7) is satisfied. Find the energy eigenvalue En. Knowing that Ĥ is an observable, what can you tell about (m|n) for m + n? 9. Using your results, show that the following formula holds: ât |n) = Vn+1 |n+1) â n) = Vn In – 1) (8a) (8b) Do this either by giving a general proof or by showing that they hold for n = = 0,1,2,3. 10. Let's go back to Eq.(2). What is the SI unit of the coefficient ? Does it make 2mw sense to you? 11. Show that the square of position operator is (âtât + â†â + â↠+ ââ) 2mw (9) 12. Compute (0|£|0) and (0|â²|0). This can be done very efficiently if you use Eq.(8) 13. Find the expression of the momentum operator square, p, in terms of the Fock oper- ators. Compute (Op|0) and (0|* |0).
The Fock operator â is defined by
(1)
||
2h
where î and p are the position and momentum operators, respectively.
1. Write dowm ât in terms of î and p.
2. Show that
(ât +â)
2mw
(2)
p=i
2
ħmw
(ât – â)
(3)
hold.
3. Show that the cannonical communation relation, [â, p = iħ, yields the so-called bosonic
commutation relation,
(â, ât] = 1.
(4)
4. Show that the Hamiltonian of the SHO,
is written as
2
2m
(*+)
Ĥ = hw Ñ +
(5)
where N = âtâ is called the number operator.
5. Show that Ñ is Hermitian. Suggestion: Use the identity from Exercise #1, (QR)† =
6. A normalized vector |0) (so that (0|0) = 1) is defined to satisfy â|0) = 0. With this
the following vectors are constructed:
|n) = A, (ât)" |0) for n =
= 0, 1,2, ...
(6)
where An are constant with A, = 1. Compute Nn) for n =
are the eigenvectors of N, i.e., Ñ|N) is proportional to |N). Find the eigenvalues of
N from the proportionality.
0, 1,2,3 to show that these
CS Scanned with CamScanner
Transcribed Image Text:The Fock operator â is defined by (1) || 2h where î and p are the position and momentum operators, respectively. 1. Write dowm ât in terms of î and p. 2. Show that (ât +â) 2mw (2) p=i 2 ħmw (ât – â) (3) hold. 3. Show that the cannonical communation relation, [â, p = iħ, yields the so-called bosonic commutation relation, (â, ât] = 1. (4) 4. Show that the Hamiltonian of the SHO, is written as 2 2m (*+) Ĥ = hw Ñ + (5) where N = âtâ is called the number operator. 5. Show that Ñ is Hermitian. Suggestion: Use the identity from Exercise #1, (QR)† = 6. A normalized vector |0) (so that (0|0) = 1) is defined to satisfy â|0) = 0. With this the following vectors are constructed: |n) = A, (ât)" |0) for n = = 0, 1,2, ... (6) where An are constant with A, = 1. Compute Nn) for n = are the eigenvectors of N, i.e., Ñ|N) is proportional to |N). Find the eigenvalues of N from the proportionality. 0, 1,2,3 to show that these CS Scanned with CamScanner
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Let denote the Fock operator. Now write the position operator in terms of Fock operator.

 

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