Intermediate Quantum Mechanics PHYS307 Professor Scott Heinekamp

Goals of the course• by speculating on possible analogies between waves moving in a uniform medium and the so-called free particle, to develop some calculational tools for describing matter waves, including the de Broglie wavelength for a moving particle, and the Born interpretion of the wave function• to ‘derive’ the Schrödinger equation(s) for said wave function for a particle in (or not in) a potential V(x)• to discuss (review?) several important potential energy cases• to explore the alternative methodology of Heisenberg’s operator algebra for the case of the harmonic oscillator potential• to work in three dimensions, and solve problems of practical importance, including the hydrogen atom• to introduce the quantum mechanical treatment of spin and orbital angular momentum• to briefly apply these ideas to many-body systems

The Spectrum of Hydrogen• bright-line (emission) spectrum: hot glowing sample of H emits light• dark-line (absorption) spectrum: cool sample of H removes light• in the visible, one sees only the Balmer series, with wavelengths given by the famous Rydberg formula (n = 3,4,5…)

• it is a miracle that we can only SEE the Balmer series• the other series are given by

nm 4

6.364nm 1

4

101097.

12

21

2

n

n

n

122

nm 11

01097.1

if nn

• Lyman: nf = 1 (all UV)

• Paschen: nf = 3 (all IR)

Explaining this result by quantizing something I• we assume that the orbits of the electrons are quantized, in the sense that if an orbiting electron in ‘orbit level’ n absorbs a photon of the correct energy, it may be ‘kicked’ all the way off to ∞• classical orbit theory: equate Coulomb force to centripetal force for an atom of atomic number Z with only one electron left on it, to get KE ( is reduced mass, which is almost the electron mass but slightly less):

• assuming a circular orbit of radius r, both PE and KE are constants

r

ZevKE

r

v

r

eZe 2

0

22

20 8

1

2 yields which

4

1

• more classical theory: r

ZePE

2

04

1

r

Ze

r

Ze

r

Ze KEPE E

2

0

2

0

2

0 8

1

8

1

4

1

[I] 1

4

)(get also we0

2

r

Zerv

Explaining this result by quantizing something II• Einstein explained the photoelectric effect by arguing that light’s energy is proportional to its frequency, and that light can only be emitted or absorbed in ‘packets’ (quanta) now called photons: E = hf• h is Planck’s constant: h = 6.626 x10–34 J∙s = 4.136 x10–15 eV∙s • incidentally, we often use ‘hbar’: ħ:=h/2 = 1.046 x10–34 J∙s• we assume that the energy to ionize requires a photon whose frequency f is half of the orbital frequency of the ‘starting’ state n, times n:

• so, we equate |E| to ½ nhforb:

• orbital frequency is forb:

23

03

2

0

2

orb1

16

11

42

1

2ncecircumfere

speed1

r

Ze

rr

Ze

r

v

Tf

or

• [Kepler’s third law: (period)2 ~ (radius)3]

[II] 2228

1

28

1

0

22

0

orb2

0 nh

Zev

r

vnh

r

Zenhf

r

ZeE n

Explaining this result by quantizing something III

• so the orbital radii are quantized… as are the orbital speeds… as are the energies of the orbits!

eV 6.138

where

11

88

1

8

220

4

0

202

2220

42

220

2

0

2

0

2

h

eE

nEZ

nh

eZ

nh

ZeZe

r

ZeE

nn

• one can show that angular momentum is quantized: L = nħ• this is equivalent to n de Broglie wavelengths around the orbit circumference

• now connect all of this together by relating the radius of the orbit to n: take the expression from [I] for v(r) and the expression from [II] for vn

and equate the two:

nm053. where:

4

1

4

2

1

4

2

20

0

202

2

20

2220

42

0

2

0

2

0

2

e

ha

Z

nan

Ze

hr

hn

eZ

r

Ze

nh

Ze

r

Ze

n

The ‘old’ theory of the hydrogen-like atom à la Niels Bohr

• it misses completely the angular dependence of ‘where’ the electron is, and it oversimplifies greatly the radial position• the electrons DO NOT ‘orbit’… they are ‘everywhere’ at once• still, the theory was a smashing success and earned a Nobel Prize

• electron energies En = – Z2 E0 n– 2 and that is very good!

• they crowd closer and closer together and there are an infinite number of them ionization at zero energy• the speeds get smaller as n goes up ~ n– 1… that’s sort of OK• the radii get larger as n goes up ~ n2… that’s sort of not so OK• in a transition from ni to nf, a photon is emitted or absorbed

whose energy is precisely the difference in the electron’s energy

1232

0

4

H22H2 nm 10 x 1.097

8 is Rydberg the where

111

ch

eR

nnRZ

if