Thursday, September 17, 2020

Solutions To Angular Momentum Problems

 1) Write each of the angular momentum operators:’

L x op   =  -i h   [y  (/x ) – x (/y)]
L y op   =  -i h   [z  (/x ) – x (/z)]
L z op   =  -i h   [x  (/y ) – y (/x)]

In spherical coordinates.

Ans.

L x op   =    iħ  [sin  f ( / q )  +  cot  q cos f   ( / f)]

L y op   =   iħ  [- cos  f ( / q )  +  cot  q sin  f   ( / f)]

L z op   =   -  iħ  ( / f)


2)   Consider the form:   [H, L x ]

Show that [H, L x ] = yZ  - zY and give the condition for H  and L x   to commute.

Soln.

Hamiltonian H  = p 2/ 2m  + V(r)

[H, L x ] = [  p 2/ 2m  + V(r)  ,   L x  ]  =  1/ 2m [ p 2 , L x  ]  + [V(r)  ,  L x ]  

[p 2 , L x  ] =  [ p x L x  ]  + [ p y L x  ]  +  p z L x  ]


 p x L x  ]  =   x  p x  L x  ]   +  p x  L x  ] p x


 p x  L x  ]  =  p x   y p z   - z p ]  =  0  

And, for constant angular momentum:  

p x  L x  ]   =  0   =  [H, L x ]  

Then:  [H, L x ] = yZ  - zY   

REM:

Z = z   = -i h   (/z )   and Y =  y   = -i h   (/y )

 H and L x  do commute ([H, L x ] = 0)  since a key property of central potential problems is that the angular momentum operators commute with the Hamiltonian .

Then:  [L x , H] =   0,   And  [H, L x ] = 0

i.e.  For all central potential Hamiltonians: [^L i,  H ] = 0 =   [H, ^L i 

 The above commutator implies that the  ^L i  operators are conserved in central potentials. 

3)  (a)

 (L’ x -  i L’ y) =

(0...0)
    (1....0)

And:  (L’ x +   i L’ y) =

(0...1)
    (0...0)

Hence the matrices are found to be Hermitian, which means the matrix is equal to its conjugate transpose.


b)  If ℓ  =  1   calculate (L’ x + i L’ y) m m’   


Soln.

(L’ x + i L’ y) m m’   =

ħ  [ ( -  m’ ) ( +  m’ + 1 ) ]1/2  exp (i m ℓ j)  d m m’+1  

=  ħ  [ (1  -  0 ) (1 +  0 + 1 ) ]1/2  exp (i 0  j)  [1]

 ħ   [Ö2 ] 

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