Transcript Introduction to Quantum Mechanics AEP3610 Professor Scott

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…)
 n2 
1
 1 1  1
 nm
 .01097  2  nm    364.6 2
 n 4

4 n 


• it is a miracle that we can
only SEE the Balmer series
• the other series are given by
 1

1
 .01097 2  2  nm 1
n


n
f
i


1
• 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 (m is
reduced mass, which is almost the electron mass but slightly less):
Zee  mv 2
mv 2
Ze2
which yields KE 

2
40 r
r
2
80 r
Ze2
1
 we also get v(r ) 
[I]
m 40 r
1 Ze2
PE  
• more classical theory:
40 r
• assuming a circular orbit of radius r, both PE and KE are constants
1

1
Ze2
1 Ze2
1 Ze2
E  PE  KE  


40 r
80 r
80 r
1
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:
• orbital frequency is forb:
1
speed
v
1
f orb 



Tor circumfere nce 2r 2
Ze2
m 40
• [Kepler’s third law: (period)2 ~ (radius)3]
• so, we equate |E| to ½ nhforb:

11

r r
Ze2
m16 3 0 r 3 2
Ze2 nhforb
1 Ze2 nh  v 
Ze2
E


 
[II]
  vn 
80 r
2
80 r
2  2r 
2 0 nh
1
1
Explaining this result by quantizing something III
• 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:
Ze2
1
Ze2
Ze2 1
Z 2e 4

m 40
r

2 0 nh

m 40 r

4 02 n 2 h 2
 0 h 2 2 a0 n 2
 0h2
 rn 
n :
where a0 
 .053 nm
2
2
Z
mZe
me
• so the orbital radii are quantized… as are the orbital
speeds… as are the energies of the orbits!
Ze2 1
Ze2 Ze2
Z 2e 4 1
1
2
 En  





Z
E
0 2
2 2 2
80 rn
80  0 h 2 n 2
80 h n
n
where E0 
e4
802 h 2
 13.6 eV
• one can show that angular momentum is quantized: L = nħ
• this is equivalent to n de Broglie wavelengths around the
orbit circumference
The ‘old’ theory of the hydrogen-like atom à la Niels Bohr
• 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
4
 1

1
m
e
 Z RH  2  2  where the Rydberg is RH  2 3  1.097 x 102 nm 1
n


8 0 h c
 f ni 
1
2
• 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