The Fock operator à is defined by (1) where i and p are the position and momentum operators, respectively. a Wite down a' in terms of à and p. b. Show that (a + a) 2mw (2) hma (a' – a) (3) hold. c. Show that the cannonical communation relation, [ż, p) = ih, yiekds the so-called bosonic commutation relation, (4)
The Fock operator à is defined by (1) where i and p are the position and momentum operators, respectively. a Wite down a' in terms of à and p. b. Show that (a + a) 2mw (2) hma (a' – a) (3) hold. c. Show that the cannonical communation relation, [ż, p) = ih, yiekds the so-called bosonic commutation relation, (4)
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Please do A, B, and C
![Exercise # 4 (updated)
The Fock operator à is defined by
(1)
where i and p are the position and momentum operators, respectively.
a. Write down ât in terms of î and p.
b. Show that
i = V a' +à)
(2)
p= i (â' – à)
(3)
hold.
c. Show that the cannonical communation relation, [2, p) = ih, yields the so-called bosonic
commutation relation,
(â, ât] = 1.
(4)
P. mu
d. Show that the Hamiltonian of the SHO, H =
2m
is written as
l = h
(5)
where N = âtâ is called the number operator.
Show that N is Hermitian. Suggestion: Use the identity from Exercise #1, (QR) =
e.
f.
A normalized vector |0) (so that (0|0) = 1) is defined to satisfy à 0) = 0. With this
the following vectors are constructed:
|n) = A, (â')" |0) for n = 0,1,2,-..
(6)
where A, are constant with Ap = 1. Compute Nn) for n = 0, 1,2,3 to show that these
are the eigenvectors of N, i.e., N|N) is proportional to |N). Find the eigenvalues of
Ñ from the proportionality.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa4426dc-92c3-4ac7-bd5a-9ba1dd10b9af%2Fd8a3aebb-dc8e-4e28-ab08-abd84e36fbe2%2F3uqu91h_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise # 4 (updated)
The Fock operator à is defined by
(1)
where i and p are the position and momentum operators, respectively.
a. Write down ât in terms of î and p.
b. Show that
i = V a' +à)
(2)
p= i (â' – à)
(3)
hold.
c. Show that the cannonical communation relation, [2, p) = ih, yields the so-called bosonic
commutation relation,
(â, ât] = 1.
(4)
P. mu
d. Show that the Hamiltonian of the SHO, H =
2m
is written as
l = h
(5)
where N = âtâ is called the number operator.
Show that N is Hermitian. Suggestion: Use the identity from Exercise #1, (QR) =
e.
f.
A normalized vector |0) (so that (0|0) = 1) is defined to satisfy à 0) = 0. With this
the following vectors are constructed:
|n) = A, (â')" |0) for n = 0,1,2,-..
(6)
where A, are constant with Ap = 1. Compute Nn) for n = 0, 1,2,3 to show that these
are the eigenvectors of N, i.e., N|N) is proportional to |N). Find the eigenvalues of
Ñ from the proportionality.
Expert Solution
Step 1
a.
b. We have,
Now, adding,
Again,
Subtracting we have:
Step by step
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