j. Let's go back to Eq (2). What is the SI unit of the coeficient ? Does it make 2mw sense to you? k. Show that the square of position operator is (a'a' + â'à + à⪠+ â) 2m (9) 1. Compute (0|#|0) and (0|2²|0). This can be done very efficiently if you use Eq-(8)
j. Let's go back to Eq (2). What is the SI unit of the coeficient ? Does it make 2mw sense to you? k. Show that the square of position operator is (a'a' + â'à + à⪠+ â) 2m (9) 1. Compute (0|#|0) and (0|2²|0). This can be done very efficiently if you use Eq-(8)
Related questions
Question
Please do J, K, and L
Transcribed Image Text:g Find A, by normalizing n). Suggestion: Compute (n|n) for n = 0, 1, 2, 3 then find the
pattern. Use the identity (Qflg) = (fIQ" |g) for Q = â'.
h. Show that n) are the eigenvectors of H, i.e.,
Ĥ \n) = E, \n)
(7)
is satisfied. Find the energy eigenvalue E. Knowing that H is an observable, what
can you tell about (m|n) for m + n?
i.
Using your results, show that the following formula holds:
à* In) = Vn +1 |n + 1)
à \n) = Vĩ \n – 1)
(8a)
(8b)
Do this either by giving a general proof or by showing that they hold for n = 0,1,2,3.
i. Let's go back to Eq.(2). What is the SI unit of the coefficient
? Does it make
2mw
sense to you?
k. Show that the square of position operator is
* =(a'â* +â'â + â⪠+ ââ)
(9)
2mu
1 Compute (0|&|0) and (0|10). This can be done very efficiently if you use Eq.(8)
Find the expression of the momentum operator square, , in terms of the Fock oper-
m
ators.
Compute (0|p|0) and (0|i²|0).
n.
![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%2F3a7ee0e2-7550-4e33-8391-5f1107cecde5%2F3ohhbb_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 (j)
According to equation (2)
The unit of Planck's constant is , the unit of mass is kg, and the unit of angular frequency is . So the SI unit of the coefficient will be:
Since the rising and lowering operators and are unitless and the unit of the coefficient is meters therefore the unit of the position operator is meter, which is the expected result.
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