Looking At Angular Momentum Operators

 As we've seen in previous posts, operators play a key role in physics.   For example, interested readers can refer to my 2014 post,

• Introduction to Quantum Mechanical Operators (Pt.1)

Where I noted:

"The brilliance of the early quantum mechanicians lay in substituting the operators ( e.g E_op, p^ or p _op  ) for the corresponding quantities of the original classical Hamiltonian, then multiplying through by the wave function y :    

H _op y  =  E_op  y

In this way, a drastically simplified basic quantum mechanical equation could be obtained, which could then be expanded once one substituted the operators, i.e.:

p _op  =  -i h (¶/ ¶ x)

E_op  =  i h  (¶ /¶t)

Other (3D system) operators  can be seen in the quantum mechanical equation:

T _op y  =  E_op  y

Where T _op is the kinetic energy operator which is written in 3D as:

T _op   =    p^ p^ / 2m  =  [p^ _x ²  +  p^ _y ²    + p^ _z ²  ] / 2m

=  - ħ²/ 2m [ ¶² / ¶ x²  +   ¶² / ¶ y²  + ¶² / ¶ z² ) = - ħ²/ 2m Ñ ²

Where Ñ is the Laplacian operator.

Angular momentum operators are also important in quantum mechanics. The physical significance of the angular momentum quantum number (ℓ) is to convey the shape of the probability density cloud or orbital for a given atom. The numbering rule for ℓ is directly contingent on the value for n. Thus, for any given n, then ℓ must be such that it has integral values from 0 to (n -1). This means if n = 2, then ℓ can have (n- 1) = (2 -1) = 1. But if n =1, then: ℓ = (n – 1) =  0.

We now look at the total angular momentum, which would be:

L = [ℓ (ℓ + 1)] ^1/2 (ħ)

Consider now  the commutator relations for the respective components  L _x    and  L _y   :

[L _x  ,   L _y   ] =  [y p _x   - zp _y ,   zp _x  - x p _z ]

=   ħ / i    =  [y p _x   - xp _y ,]  =  i ħ L _z

Thus:   [L _x ,   L _y ]  ≠    0

From classical mechanics we saw for the angular momentum :

ℓ  =  r   x  p  =  r p _q  =  r p sin q 

Where r and p are the magnitudes of the position of the particle relative to the origin and its momentum, respectively.  But in quantum mechanics, say for rectilinear coordinates ,we have:

         L =   

   

 -i h      [i                j                  k ]

             [x             y                    z]

             [¶/¶x        ¶/¶y       ¶/¶ z]

Where:     L _z  =    -i h [x  ¶/¶y   -  y ¶/¶x  ]         

And as before  we use L to distinguish the total angular momentum from the angular momentum quantum number ℓ since the magnitude of L is: L = (ħ)   Ö [ℓ (ℓ + 1)]

It is straightforward to show the classical rectangular components of L  are:

L _x  =  y p _x   - z p _y

L _y  =  z p _x   - xp _z

L _z  =  x p _y   - y p _x

Where x, y, z are the components of r, and  p _x , p _y  and  p _z are the components of p.

In quantum mechanics the angular momentum operators are obtained by replacing the momentum components from the classical case by their quantum mechanical equivalents, i.e.

p _x    =  -i h   (¶/¶x )

p _y    =  -i h   (¶/¶y )

p _z    =  -i h   (¶/¶z )

The quantum mechanical angular momentum operators are then written:

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)]

It is also useful here to note the role of the Pauli spin matrices  (s  _x   etc.), e.g.

L’ _x   =  h /2   ( s  _x )

L’ _y   =  h /2   ( s  _y )

L’ _z   =  h /2   ( s  _z )

Where:

s  _x     =   

(0.......1)
(1.......0);

s  _y     =   

(0.......-i)
(i.........0)

s  _z     =   

(1.......0)
(0......-1)

And we note here each of these matrices is Hermitian.  

(See e.g.   Hermitian Matrices and Orbital Angular Momentum  )

Further, since  L’ _x    and   L’ _y  are Hermitian  then (L’ _x + i L’ _y) and  (L’ _x  -   i L’ _y) are Hermitian conjugates.

It can also be shown:

L _x L _y    =

 - h  ²  (y  ¶/¶z ) – (z  ¶/¶y )( z  ¶/¶x  – x ¶/¶z)

And:

[L, L _y  ] =   i h  ²  L _z

[ L _y   , L _z ] =   i h  ²  L _x

[ L _z   , L _x ] =   i h  ²  L _y

i.e. none of the components of angular momentum commute with each other.

Further:

[ L _x   , L _y ] L _y  =   i h  L _y    L _z

And:

L _y   [ L _x   , L _y ] =   i h  L _y    L _z

_{Problems for the Physics enthusiast:}

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.

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.

3) (a)  Show that (L’ _x + i L’ _y)   and (L’ _x -  i L’ _y)   are Hermitian conjugates.

b) A matrix form for one of the angular momentum Hermitan conjugates may be written:

(L’ _x + i L’ _y) _{m m’}   =

ħ  [ (ℓ -  m’ ) (ℓ +  m’ + 1 ) ]^1/2  exp (i m _ℓ j)  d _{m m’+1}  

Where d _{m m’ +1}     is the Kronecker delta applied to the quantum numbers m and m’ + 1.   The table below may be used to compute the appropriate value for the case given.

If ℓ  = 1    calculate (L’ _x + i L’ _y) _{m m’}