Download Quantum Mechanics in a Nutshell

Quantum Mechanics in a Nutshell
Quantum theory
• Wave-particle duality of light (“wave”) and electrons
(“particle”)
• Many quantities are “quantized” (e.g., energy,
momentum, conductivity, magnetic moment, etc.)
• For “matter waves”: Using only three pieces of
information (electronic charge, electronic mass, Planck’s
constant), the properties of atoms, molecules and solids
can be accurately determined (in principle)!
Quantum theory – Light as particles
• Max Planck (~1900): energy of electromagnetic (EM) waves can
take on only discrete values: E = nħ
from density of states
from equipartition theorem
– Why? To fix the “ultraviolet catastrophe”
– Classically, EM energy density,   ~  2avg =  2(kT)
– But experimental results could be recovered only if energy of a mode is
an integer multiple of ħ as
å(n w )e
=
åe w
-n w / kT
e avg
n
-n
/ kT
=
w
e
w / kT
-1
n
 
The ultraviolet
catastrophe
Classical (~ 2kT)
experimental

Quantum theory – Light as particles
• Einstein (1905): photoelectric effect
– No matter how intense light is, if  <  c  no photoelectrons
– No matter how low the intensity is, if  >  c, photoelectrons result
– Light must come in packets (E = nħ )
• Compton scattering (1923): establishes that photons have
momentum!
– Scattering of x-rays of a single frequency by electrons in a graphite target
resulted in scattered x-rays
– This made sense only if the energy and the momentum were conserved, with the
momentum given by p = h/ = ħk (k = 2/ , with  being the wavelength)
•
By now, it is accepted that waves may display particle features …
Quantum theory – Electrons as waves
• Rutherford (~1911): Experiments indicate that
atoms are composed of positively charged
nuclei surrounded by a cloud of “orbiting”
electrons. But,
– Orbiting (or accelerating) charge radiates energy 
electrons should spiral into nucleus  all of matter
should be unstable!
– Spectroscopy results of H (Rydberg states) indicated
that energy of an electron in H could only be -13.6/n2
eV (n = 1,2,3,…)
Quantum theory – Electrons as waves
• Bohr (~1913):
– Postulates “stationary states” or “orbits”, allowed only if electron’s
angular momentum L is quantized by ħ, i.e., L = nħ implies that E = 13.6/n2 eV
– Proof:
• centripetal force on electron with mass m and charge e, orbiting with velocity v at
radius r is balanced by electrostatic attraction between electron and nucleus 
mv2/r = e2/(4pe0r2)  v = sqrt(e2/(4pe0mr))
• Total energy at any radius, E = 0.5mv2 - e2/(40r) = -e2/(80r)
• L = nħ  mvr = nħ  sqrt(e2mr/(40)) = nħ  allowed orbit radius, r =
40n2ħ2/(e2m) = a0n2 (this defines the Bohr radius a0 = 0.529 Å)
• Finally, E = -e2/(80r) = -(e4m/(802h2)).(1/n2) = -13.6/n2 eV
– The only non-classical concept introduced (without justification): L = nħ
Quantum theory – Electrons as waves
• de Broglie (~1923): Justification: L = nħ is equivalent to
n = 2r (i.e., circumference is integer multiple of
wavelength) if  = h/p (i.e., if we can “assign” a
wavelength to a particle as per the Compton analysis for
waves)!
– Proof: n = 2r  n(h/(mv)) = 2r  n(h/2) = mvr  nħ = L
• It all fits, if we assume that electrons are waves!
Quantum theory – Electrons as waves
The Schrodinger equation: the jewel of the crown
•
•
Schrodinger (~1925-1926): writes down “wave equation” for any single
particle that obeys the new quantum rules (not just an electron)
A “proof”, while remembering: E = ħ & p = h/ = ħk
–
–
–
For a free electron “wave” with a wave function Ψ(x,t) = ei(kx- t), energy is purely kinetic
Thus, E = p2/(2m)  ħ = ħ2k2/(2m)
A wave equation that will give this result for the choice of ei(kx- t) as the wave function is
2
¶
¶2
i
Y(x,t) = 2 Y(x,t)
¶t
2m ¶x
•
Schrodinger then “generalizes” his equation for a bound particle
é 2 ¶2
ù
¶
i
Y(x,t) = ê2 + V (x) úY(x,t)
¶t
ë 2m ¶x
û
K.E.
P.E.
Hamiltonian operator
The Schrodinger equation
• In 3-d, the time-dependent Schrodinger equation is
é 2 æ ¶2 ¶2 ¶2 ö
ù
¶
i
Y(x, y,z,t) = êç 2 + 2 + 2 ÷ + V (x, y,z)úY(x, y,z,t)
¶t
ë 2m è ¶x ¶y ¶z ø
û
• Writing Ψ(x,y,z,t) = ψ(x,y,z)w(t), we get the timeindependent Schrodinger equation
é 2 æ ¶2 ¶2 ¶2 ö
ù
êç 2 + 2 + 2 ÷ + V (x, y,z)úy (x, y,z) = Ey (x, y,z)
ë 2m è ¶x ¶y ¶z ø
û
Hamiltonian, H
• Note that E is the total energy that we seek, and
Ψ(x,y,z,t) = ψ(x,y,z)e-iEt/ħ
The Schrodinger equation
H  E
• An eigenvalue problem
– Has infinite number of solutions, with the solutions being Ei
and ψi
– The solution corresponding to the lowest Ei is the ground state
– Ei is a scalar while ψi is a vector
– The ψis are orthonormal, i.e., Int{ψi(r)ψj(r)d3r} = ij
– If H is hermitian, Ei are all real (although ψi are complex)
– Can be cast as a differential equation (Schrodinger) or a
matrix equation (Heisenberg)
– |ψ|2 is interpreted as a probability density, or charge density
Applications of 1-particle Schrodinger equation
• Initial applications
– Hydrogen atom, Harmonic oscillator, Particle in a box
• The hydrogen atom problem
é 2 æ ¶2 ¶2 ¶2 ö
ù
êç 2 + 2 + 2 ÷ + V (x, y,z)úy nlm (x, y,z) = E nlmy nlm (x, y,z)
ë 2m è ¶x ¶y ¶z ø
û
1 ¶æ 2 ¶ö
1
¶ æ
¶ ö
1
¶2
2
Ñ = 2 çr
÷+
çsin q ÷ + 2 2
r ¶r è dr ø r 2 sin q ¶q è
dq ø r sin q ¶j 2
-
e2
4pe 0 r
Solutions: Enlm = -13.6/n2 eV; ψnlm(r,θ,ϕ) = Rn(r)Ylm(θ,ϕ)
http://www.falstad.com/qmatom/
http://panda.unm.edu/Courses/Finley/P262/Hydrogen/WaveFcns.html
Summary of quantization
•
Spin (Pauli exclusion principle) not included in the Schrodinger equation &
needs to be put in by hand (but fixed by the Dirac equation)
The many-particle Schrodinger equation
• The N-electron, M-nuclei Schrodinger (eigenvalue) equation:
 (r1 , r2 ,..., rN , R1 , R2 ,..., RM )  E(r1 , r2 ,..., rN , R1 , R2 ,..., RM )
The N-electron, M-nuclei wave function
The total energy that we seek
The N-electron, M-nuclei Hamiltonian
2
2
2
2
N
M M
N N
M N
Z
Z
e
2

1
1
e
Z
e
  
 2I  
i2   I J  
  I
2 I 1 J  I RI  RJ 2 i 1 j  I ri  rj I 1 i 1 RI  ri
I 1 2 M I
i 1 2m
M
Nuclear kinetic
energy
Electronic
kinetic energy
Nuclear-nuclear
repulsion
Electron-electron
repulsion
Electron-nuclear
attraction
• The problem is completely parameter-free, but formidable!
– Cannot be solved analytically when N > 1
– Too many variables – for a 100 atom Pt cluster, the wave function is a
function of 23,000 variables!!!
The Born-Oppenheimer approximation
• Electronic mass (m) is ~1/1800 times that of a nucleon mass (MI)
• Hence, nuclear degrees of freedom may be factored out
• For a fixed configuration of nuclei, nuclear kinetic energy is zero
and nuclear-nuclear repulsion is a constant; thus
H elec (r1 , r2 ,..., rN )  Eelec (r1 , r2 ,..., rN )
M N
 2 2 1 N N e2
Z I e2
 
i  
 
2 i 1 j  I ri  rj I 1 i 1 RI  ri
i 1 2m
N
H elec
1 M M Z I Z J e2
E  Eelec  
2 I 1 J  I RI  RJ
Electronic eigenvalue problem is still difficult to solve!
Can this be done numerically though? That is, what if we chose a known
functional form for ψ in terms of a set of adjustable parameters, and figure out a
way of determining these parameters?
In comes the variational theorem
The variational theorem
• Casts the electronic eigenvalue problem into a minimization problem
• Lets introduce the Dirac notation
 (r1 , r2 ,..., rN )  
H elec   Eelec 
*
3
3
3
...

(
r
,
r
,...,
r
)

(
r
,
r
,...,
r
)
d
r
d
r
...
d
   1 2 N 1 2 N 1 2 rN   
*
3
3
3
...

(
r
,
r
,...,
r
)
H

(
r
,
r
,...,
r
)
d
r
d
r
...
d
rN   H 
1 2
N
1
2
  1 2 N
• Note that the above eigenvalue equation has infinite solutions: E0, E1, E2,
… & correspondingly ψ0, ψ1, ψ2, …
• Our goal is to find the ground state (i.e., the lowest energy state)
• Variational theorem
– choose any normalized function F containing adjustable parameters, and
determine the parameters that minimize <F|Helec|F>
– The absolute minimum of <F|Helec|F> will occur when F = ψ0
– Note that E0 = <ψ0|Helec|ψ0> thus, strategy available to solve our problem!
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