d. Show that the Hamiltonian of the SHO, H = 2m is written as - hwN+ (5) where N- ala is called the mmber operator. е. Show that N is Hermitian. Suggostion: Use the idlentity from Exercise #1, (QR)' = f. A normalized vector |0) (so that (0/0) = 1) is defined to satisfy å|0) = 0. With this the following veetors are constructed: In) = A, (a')" (0) for n = 0, 1,2, . (6) where A, are constant with A, 1. Compute Ñ\n) for n = 0, 1, 2, 3 to show that these are the cigenvectors of N, i.e., Ñ|N) is proportional to |N). Find the eigenvalues of N from the proportionality.
d. Show that the Hamiltonian of the SHO, H = 2m is written as - hwN+ (5) where N- ala is called the mmber operator. е. Show that N is Hermitian. Suggostion: Use the idlentity from Exercise #1, (QR)' = f. A normalized vector |0) (so that (0/0) = 1) is defined to satisfy å|0) = 0. With this the following veetors are constructed: In) = A, (a')" (0) for n = 0, 1,2, . (6) where A, are constant with A, 1. Compute Ñ\n) for n = 0, 1, 2, 3 to show that these are the cigenvectors of N, i.e., Ñ|N) is proportional to |N). Find the eigenvalues of N from the proportionality.
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Please do D, E, and F
![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%2Fc6a217a2-0a0b-4c4a-b5af-6c060c971b1c%2Fm6klip_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.
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