1Position OperatorLet : R C be a 1D wavefunction; we define the position operator as:

x = x

If we are in three dimensions, well have:~r = xe1 + ye2 + ze3~r = xe1 + ye2 + ze3

Expressing everything in spherical coordinates:~r = e

The eigenfunction for the Position Operator is the delta functions cente-red at x0 with eigenvalue x0:

x(x x0) = x0(x x0)

Momentum OperatorLet : R C be a 1D wavefunction; we define the momentum operator

as:

p = i~x

If we are in three dimensions, well have:~p = i~

The eigenfunction for the Momentum Operator are:

p = p (1)

i~x

= p (2)

x=

ip

~ (3)

Setting p = ~k we have:(x) = Aeikx = A(cos kx+ i sin kx)

2Commutations Relations: Position-Momentun

In one dimension we have:

[x, p] = i~Proof:

x(p) p(x) = x(i~)x (i~)(x)

x= i~(x

x x

x) = i~

In three dimensions we have:

[xi, pj] = i~ijFor example:

[x, px] = i~[x, py] = 0

[x, pz] = 0

Orbital Angular Momentum Operator

Like in classical mechanics, we define the angular momentum as:

~L = ~r ~p = i~~r ~

In cartesian coordinates we have:~L = (Lx, Ly, Lz) = i~[(y z z y )e1 + (z x x z )e2 + (x y y x)e3]In spherical:

~L = i~(e 1sin e )

L2 = ~2sin2

2

2 ~2

sin

(sin

)

Lx = i~(sin + cos cot

)

Ly = i~( cos + sin cot )L = ~ei( + i cot )Lz = i~ Useful relations:

[Lx, Ly] = i~Lz [Ly, Lz] = i~Lx [Lz, Lx] = i~Ly[L2, Lx] = 0 [L

2, Ly] = 0 [L2, Lz] = 0

Since L2 and Lz commute, we can find common eigenfunctions for bothoperators; lets call the Ylm

For Lz we have (measuring eigenvalues in units of ~ ):

LzYlm = ~mYlm

3i~Ylm

= ~mYlm

Ylm = P ()eim

where P is a function of only variableLets impose some conditions:

Ylm(, 0) = Ylm(, 2pi)

P ()eim(0) = P ()eim(2pi)

m N

Now solve for L2:

L2Ylm = ~2

sin2

2Ylm2

~2

sin

(sin

Ylm

)

L2(P ()eim) = ~2

sin2

2(P ()eim)

2 ~

2

sin

(sin

(P ()eim)

)

eim~2l(l + 1)P () = ~2

sin2 P ()(m2)eim eim ~

2

sin

(sin

P ()

)

P ()[m2

sin2 l(l + 1)] = 1

sin

(sin

P ()

)

Setting x = cos we obtain:(1 x2)P (x) 2xP + [l(l + 1) m2

1x2 ]P = 0