Physics 451

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

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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

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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

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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

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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

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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

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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