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 ² , L _x  ]  + [ p _y ² , L _x  ]  +  [ p _z ² , L _x  ]

 [ p _x ² , L _x  ]  =   p _x  [ p _x  , L _x  ]   +  [ p _x  , L _x  ] p _x

 _{[ p x  , L x  ]  =}  [ p _x  ,  y p _z   - z p _y ]  =  0  

And, for constant angular momentum:  

[ p _x  , L _x  ]   =  0   =  [H, L _x ]  

Then:  [H, L _x ] = yZ  - zY   

REM:

Z = p _z   = -i h   (¶/¶z )   and Y =  p _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) =

h  (0...0)
    (1....0)

And:  (L’ _x +   i L’ _y) =

h  (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  [ (ℓ -  m’ ) (ℓ +  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 ]