Download Quantum Mechanics Lecture 24 Dr. Mauro Ferreira

Quantum Mechanics
Lecture 24
Dr. Mauro Ferreira
E-mail: [email protected]
Room 2.49, Lloyd Institute
Eigenfunctions
We need to rewrite Lx, Ly and Lz in spherical coordinates:
!ˆ !
ˆ
!
L = !r × ∇
i
!
ˆ
!
⇒ L=
i
!
where
∂
1 ∂
1
∂
!
∇ = r̂
+ θ̂
+ φ̂
∂r
r ∂θ
r sin θ ∂φ
∂
1 ∂
φ̂
− θ̂
∂θ
sin θ ∂φ
"
Rewriting the unit vectors θ̂ and φ̂ in spherical coordinates, we have
θ̂ = (cos θ cos φ) î + (cos θ sin φ) ĵ − (sin θ) k̂
φ̂ = − (sin φ) î + (cos φ) ĵ
!
!
∂
∂
L̂x =
− sin φ
− cos φ cot θ
i
∂θ
∂φ
!
"
!
∂
∂
L̂y =
cos φ
− sin φ cot θ
i
∂θ
∂φ
! ∂
L̂z =
i ∂φ
"
The raising and lowering operators become
L̂± = ±! e
±iφ
L̂+ L̂− = −!2
!
!
∂
∂
± i cot θ
∂θ
∂φ
"
2
∂2
∂
∂
∂
2
+ cot θ
+ cot θ 2 + i
∂θ2
∂θ
∂φ
∂φ
"
Finally, the operator L̂2 is written as
L̂ = −!
2
2
!
1 ∂
sin θ ∂θ
"
∂
sin θ
∂θ
#
1
∂
+
sin2 θ ∂φ2
2
$
The eigenvalue equations for L̂2 and L̂z become
2
L̂
f!m
= −!
L̂z f!m
2
!
1 ∂
sin θ ∂θ
"
∂
sin θ
∂θ
! ∂ m
=
f! = ! m f!m
i ∂φ
#
1
∂
+
sin2 θ ∂φ2
2
$
f!m = !2 $($ + 1) f!m
Remember lecture 18 ?
In general V (x, y, z) != V (x) + V (y) + V (z)
Typically V (!r) = V (r) (central potentials)
(Laplacian in spherical coordinates)
∇2 =
1 ∂
r2 ∂r
!
r2
∂
∂r
"
+
1
∂
r2 sin θ ∂θ
!
sin θ
∂
∂θ
"
1
r2 sin2 θ
+
!
2
∂
∂φ2
"
(Schroedinger equation in spherical coordinates)
!
!
1 ∂
−
2m r2 ∂r
2
"
r
2 ∂ψ
∂r
#
1
∂
+ 2
r sin θ ∂θ
"
∂ψ
sin θ
∂θ
#
1
+ 2
r sin2 θ
"
2
∂ ψ
∂φ2
#$
+ V ψ = Eψ
attempted solution: ψ(r, θ, φ) = R(r) Y (θ, φ)
−
!
!
Y ∂
2m r2 ∂r
2
"
r2
∂R
∂r
#
+
R
∂
r2 sin θ ∂θ
"
sin θ
∂Y
∂θ
#
+
R
r2 sin2 θ
2
−2mr
dividing by R Y an multiplying by
!2
"
2
∂ Y
∂φ2
#$
+ V RY = ERY
Lecture 18
!
1 d
R dr
"
r
2 dR
1
+
Y
dr
!
#
$
2mr
−
[V (r) − E]
!2
1 ∂
sin θ ∂θ
2
"
∂Y
sin θ
∂θ
#
function of r only
1 ∂ Y
+
sin2 θ ∂φ2
2
$
=0
function of θ and Φ only
Separating the variables...
(Radial ODE)
!
"
1 d
2mr2
2 dR
r
−
[V (r) − E] = !(! + 1)
2
R dr
dr
!
1
Y
!
(Angular PDE)
1 ∂
sin θ ∂θ
"
∂Y
sin θ
∂θ
#
1 ∂ Y
+
sin2 θ ∂φ2
2
$
separation constant
= −$($ + 1)
Lecture 18
Angular equation
1
Y
!
1 ∂
sin θ ∂θ
"
∂Y
sin θ
∂θ
#
1 ∂ Y
+
sin2 θ ∂φ2
2
$
= −$($ + 1)
multiplying by Y sin2 θ
∂
sin θ
∂θ
!
∂Y
sin θ
∂θ
"
∂2Y
2
+
=
−$($
+
1)
sin
θY
∂φ2
Separating the variables again: Y (θ, φ) = Θ(θ) Φ(φ)
!
"
#
$%
&
1
d
dΘ
1 d2 Φ
2
sin θ
sin θ
+ "(" + 1) sin θ +
=0
Θ
dθ
dθ
Φ dφ2
!
"
#$
1
d
dΘ
sin θ
sin θ
+ "(" + 1) sin2 θ = m2
Θ
dθ
dθ
1 d2 Φ
2
=
−m
Φ dφ2
Lecture 18
2
L̂
f!m
= −!
L̂z f!m
1
Y
!
2
!
1 ∂
sin θ ∂θ
"
∂
sin θ
∂θ
#
1
∂
+
sin2 θ ∂φ2
2
$
f!m = !2 $($ + 1) f!m
! ∂ m
=
f! = ! m f!m
i ∂φ
(Angular PDE)
1 ∂
sin θ ∂θ
"
∂Y
sin θ
∂θ
#
1 ∂ Y
+
sin2 θ ∂φ2
2
$
= −$($ + 1)
1 d2 Φ
2
=
−m
Φ dφ2
A simple comparison indicates that the eigenfunctions f!m are given
by the spherical harmonics Y!m (θ, φ).
CONCLUSION: Spherical harmonics are eigenfunctions of the
operators L̂2 and L̂z .
When solving the Schroedinger equation by separation of variables we
were inadvertently building simultaneous eigenfunctions of the three
commuting operators Ĥ, L̂2 and L̂z .
(Schroedinger equation in spherical coordinates)
!
!
1 ∂
−
2m r2 ∂r
2
"
r
2 ∂ψ
∂r
#
1
∂
+ 2
r sin θ ∂θ
can be simplified to
"
∂ψ
sin θ
∂θ
#
1
+ 2
r sin2 θ
"
2
∂ ψ
∂φ2
#$
+ V ψ = Eψ
!
"
#
$
1
2 ∂
2 ∂
2
−!
r
+
L̂
ψ+V ψ =Eψ
2
2mr
∂r
∂r
The only discrepancy is in the fact that the quantum numbers !
associated with eigenvalues of the angular momentum L̂2 may be
integer or half-integer whereas they were found to be integers by
solving the Schroedinger Eq. by separation of variables.
Spin angular momentum
Classically, a particle possess orbital and
! =L
!o + L
!s
spin angular momenta, i.e., L
In QM, in addition to their orbital angular momentum,
ˆ
!
elementary particles also carry an intrinsic momentum S
called spin (in analogy with classical mechanics)
Angular momentum commutation relations are still valid:
[Ŝx , Ŝy ] = i! Ŝz ; [Ŝy , Ŝz ] = i! Ŝx ; [Ŝz , Ŝx ] = i! Ŝy
Ŝ 2 |s, ms ! = !2 s(s + 1) |s, ms !
Ŝz |s, ms ! = ! ms |s, ms !
Ŝ± |s, ms ! = !
!
Defining Ŝ± ≡ Ŝx ± iŜy
s(s + 1) − ms (ms ± 1) |s, ms ± 1!
!s
L
The allowed values for the quantum numbers s and ms are
1
3
s = 0, , 1, , ...
2
2
and
ms = −s, −s + 1, ..., s − 1, s
Unlike the orbital degree of freedom, the spin of an elementary particle
is fixed, that is, s is constant. One very special case is for s=1/2, in
which case there are only two eigenstates:
1
1
and ms = −
ms = +
2
2
1 1
1 1
| , + ! and | , − "
2 2
2 2
1
1
| , ↑" and | , ↓"
2
2
| ↑ " and | ↓ "
Using these as a basis, the general state
of a spin-1/2 particle can be written as
a linear combination
|χ ! = a | ↑ ! + b | ↓ !