1. Quantum mechanics I Fall 2012
Physics 451
Quantum mechanics I
Fall 2012
Nov 2, 2012
Karine Chesnel

2. Homework tonight: HW #16 Friday Nov 2 by 7pm Homework next week:
Phys 451
Announcements
Homework tonight:
HW #16 Friday Nov 2 by 7pm
Homework next week:
HW #17 Tuesday Nov 6 by 7pm
HW #18 Thursday Nov 8 by 7pm

3. Phys 451
Normalization z r y x
Radial part or
Angular part

4. Orthonormality Pb 4.6 Phys 451 Spherical harmonics are orthogonal yz r
Spherical harmonics are orthogonal y x
Angular part
Pb 4.6
Also

5. Schrödinger equation in spherical coordinates
Phys 451
Schrödinger equation in
spherical coordinates x y z r
The radial equation
The angular equation

6. The radial equation Phys 451 y z r x
Change of variables
Form identical to Schrödinger equation
with an effective potential
Centrifugal
term

7. The radial equation V=0 Phys 451 Example: infinite spherical well
Change of variables
Inside the well

8. Three quantum numbers: (n, l, m)
Phys 451
The radial equation
V=0
Infinite spherical well
For l = 0
solution
Boundary condition
Three quantum numbers: (n, l, m)
here

9. The radial equation V=0 Phys 451 Infinite spherical well If l ≠ 0
Physical condition at r =0
solution
Spherical
Bessel function
Spherical
Neumann function

10. The radial equation V=0 Phys 451 Infinite spherical well If l ≠ 0
Physical condition at r = a :

11. Quiz 22 Phys 451 When a particle is subject to a potential
that depends on the radius only,
which quantum numbers apply to quantize the energy?
A. Only the principal quantum number n
B. Only the azimuthal quantum number l
C. Only the magnetic quantum number m
D. Possibly both numbers (n,l)
E. Possibly all three numbers (n,l,m)

12. Pb 4.9: Finite spherical well
Phys 451
V=0
Spherical well
Pb 4.7: construct and
show that they blow up at zero
Pb 4.8: case of l = 1
show that
Pb 4.9: Finite spherical well
V=-V0
Find the ground state (l =0)