physics 451
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Physics 451. Quantum mechanics I Fall 2012. Nov 9, 2012 Karine Chesnel. Phys 451. Announcements. HW #18 today Nov 9 by 7pm Homework next week: HW #19 Tuesday Nov 13 by 7pm HW #20 Thursday Nov 15 by 7pm. Step1 : determine the principal quantum number n. - PowerPoint PPT PresentationTRANSCRIPT
Physics 451
Quantum mechanics I
Fall 2012
Nov 9, 2012
Karine Chesnel
Announcements
Phys 451
•HW #18 today Nov 9 by 7pm
Homework next week:
• HW #19 Tuesday Nov 13 by 7pm
• HW #20 Thursday Nov 15 by 7pm
The hydrogen atom
How to find the stationary states?
),()(,, mlnlnlm YrRr
nakn
1Step1: determine the principal quantum number n
Step 2: set the azimuthal quantum number l (0, 1, …n-1)
Step 3: Calculate the coefficients cj in terms of c0 (from the recursion formula, at a given l and n)
Step 4: Build the radial function Rnl(r) and normalize it (value of c0)
Step 5: Multiply by the spherical harmonics (tables) and obtain 2l +1 functions nlm for given (n,l)
),( mlY
(Step 6): Eventually, include the time factor: /),,(),( tiEnlm
nertr
Phys 451
The hydrogen atom
Representation of
,,rnlm
Bohr radius
2100
2
40.529 10a m
me
Phys 451
Quantum mechanics
The hydrogen atom
Expectation values
, , ,nlm nl mlr R r Y
2 2r r R r dr22 2 2r r R r dr
2 2sin cos sinx d d r R r dr
Pb 4.13
Most probable values
Pb 4.14 2 2
maxr
2 2
0d r
dr
Quantum mechanics
The hydrogen atom
Expectation values for potential
, , ,nlm nl mlr R r Y
22 2
04
eV R r dr
r
Pb 4.15
The angular momentum
L r p
,
,
,
x y z
y z x
z x y
L L i L
L L i L
L L i L
2 2 2, , , 0x y zL L L L L L
Pb 4.19
Phys 451
The hydrogen atom
Representation of
,,rnlm
Anisotropy along Z axis
Phys 451
The angular momentum
x yL L iL
2
,
,
, 0
z
z
L L L
L L L
L L
Ladder operator
• If eigenvector of L2, then eigenvector of L2, same eigenvalueL ff
• If eigenvector of Lz with eigen value then eigenvector of Lz, new eigenvalue L f
f
Phys 451
The angular momentum
x yL L iL Ladder operator
L
L
2 2z zL L L L L
TopValue=+l
BottomValue = -l
Eigenstates m ml lf Y
2 2 ( 1)m ml lL f l l f
m mz l lL f mf
1m m ml l lL f f
Pb 4.18
Phys 451
Quiz 25
When measuring the vertical component of the angular momentum (Lz )
of the state , what will we get? 3 25L Y
A. 0
B.
C.
D.
E.
2
5
3
Phys 451
The angular momentumin spherical coordinates
1
sinL r r r r r
i r
x
y
z
r
1
sinL
i
zL i
L r p ri
Phys 451
The angular momentumIn spherical coordinates
x
y
z
r
cotiL e i
x yL L iL
22 2
2 2
1 1sin
sin sinL
Pb 4.21, 4.22
Phys 451
The angular momentumeigenvectors
x
y
z
r
m m mz l l lL f f m f
i
22 2 2
2 2
1 1sin ( 1)
sin sinm m ml l lL f f l l f
and
were the two angular equations for the spherical harmonics!
Spherical harmonicsare the
eigenfunctions
nml n nmlH E
2 2 ( 1)nml nmlL l l
z nml nmlL m
Phys 451
The angular momentumand Schrödinger equation
x
y
z
r
2 22
1
2r L V E
mr r r
3 quantum numbers (n,l,m)
• Principal quantum number n: integer• Azimutal and magnetic quantum numbers (l,m)
can also be half-integers.
Phys 451