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
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
Related questions
Question
Please answer #12 and #14. The first page is provided for context.
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b2dbe0f-1b57-45b1-b136-e64e5f1cae33%2F74358810-d2e3-4b89-92a7-7ef336552eb6%2Fbsfs7n_processed.jpeg&w=3840&q=75)
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
Expert Solution
Step 1
Hello. Since you have posted multiple questions and not specified which question needs to be solved, we will solve the first question for you. If you want any other specific question to be solved, then please resubmit only that question or specify the question number.
Let â denote the Fock operator. Now write the position operator in terms of Fock operator.
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