Physics 550 Problem Set #1 Due Wednesday Sept. 4th * Do not use Mathematica or other symbolic manipulation software on this home-work assignment. Show derivations. Finally, write in complete...

Microsoft Word - Solutions.docx
1)
a) If we have two functions F F(p,q, t) and G g(p,q, t)= = defined in canonical coordinates then poisson
acket is given by
n
i 1 i i i i
F G G F
[F,G]
q p q p=
 ∂ ∂ ∂ ∂
= − 
∂ ∂ ∂ ∂ 
∑
In our case its given that , the system having only one degree of freedom
i in 1 q q, p p and we have G x , F F(p)∴ = ⇒ = = = =
x F(p) F(p) x
[x,F(p)]
x p x p
F(p) F(p)
1 0x0 F F(p) 0
p x
∂ ∂ ∂ ∂
∴ = −
∂ ∂ ∂ ∂
∂ ∂ 
= − = ⇒ = 
∂ ∂ 
∵

Hence,
F(p)
[x,F(p)]
p
∂
=
∂

)
We expand the exponential in terms of power series and the fact that commutator is a linear operator
n
n
n 0
ˆipa 1 ia
ˆx,exp xp
n!
∞
=
      ∴ =          
∑
� �

We have ˆˆ[x,p] i= � (from definition )
We prove by mathematical induction that n n 1ˆ ˆˆ[x,p ] i np −= � for n ≥ 1
we then assume that the statement holds for n and prove that it holds for n 1+ ,
( )
( )
n 1 n 1 n 1
n n
n n n
n 1 n n
n n
n
ˆ ˆ ˆˆ[x,p ] xp p x
ˆ ˆ ˆ ˆxp p p px
ˆ ˆ ˆ ˆ ˆxp p x p p px
ˆ ˆ ˆ ˆ ˆi np p x p p px
ˆ ˆ ˆˆi np p [x, p]
ˆi (n 1)p
+ + +
−
= −
= −
 = + − 
= + −
= +
= +
�
�
�

We got the statement to be true.
Now, using the above identity , we have
( )
n
n 1
n 1
n 1
n 1
n 1
ˆipa 1 ia
ˆx,exp i np n starts from 1, n=0 term gives 0
n!
ia 1 ia
ˆ(i ) p
(n 1)!
ˆipa
a exp
∞
−
=
−∞
−
=
    
=        
   
=    
−   
 
= −   
∑
∑
� ∵
� �
�
� �
�

c)
Given that 0 0 0x̂x x x=
Let o
ˆipa
exp x
 
ψ =   �
, then we have
o
o
o o
o o
ˆipa
ˆ ˆx x exp x
ˆ ˆipa ipa
ˆ ˆexp x x,exp x
ˆ ˆipa ipa
exp x a exp x
ˆipa
(x a)exp x
 
ψ =   
     
= +          
    
= −        
 
= −   
�
� �
� �
�

Hence, o
ˆipa
exp x
 
ψ =   �
is an eigenstate of the operator x̂ with eigenvalues ( ox -a) .
2)
Hermitian operator
An operator is said to be Hermitian , if that operator is self adjoint , hence by definition
ˆ ˆA A+=
In
a-ket notation , we have i j i j
ˆ ˆA A *φ φ = φ φ
Some of the properties of Hermitian operator are
• The eigenvalue of a Hermitian operator are real
• Eigen vector co
esponding to different eigenvalues are orthogonal to each other .
• All the eigenvectors of a Hermitian operator form a complete set, that means any a
itrary wave function can
e expanded in terms of this eigenvectors
Unitary operator
An operator is said to be Unitary , if inverse of this operator is equal to its self adjoint ,
hence by definition
ˆ ˆ ˆ ˆAA A A I+ += =
Some of the properties of Hermitian operator are
• Unitary operator preserves the inner product of the Hilbert space
• Unitary operator are surjective
Orthogonal operator
A linear operator T : V V→ is orthogonal if , T(u),T(v) u, v for all u, v V= ∈
Some of the properties of orthogonal operator are
• Unitary operator preserves all the lengths and angle s.
• T(v) v for all v V= ∈
3)
i) Determinant of a unitary matrix = product of all its eigenvalues.
ii) Any real eigenvalue of an orthogonal matrix has absolute value 1 , and the determinant of an orthogonal
matrix is equal to 1 or -1
4)
Legendre’s differential equation is given by
( ) ( )21 x y 2xy n n 1 y 0, (1)′′ ′− − + + = − − − −
Where n is a given general number.
Any solution of Eq. (1) is called a Legendre function.
Dividing Eq. (1) by ( )21 ,x− one obtains
( )
2 2
n n 12x
y y y 0
1 x 1 x
+
′′ ′− + =
− −

and one observes, using the geometric series
2 4 6
2
1
1 ,
1
x x x
x
= + + + + ⋅ ⋅ ⋅
−

that the...