Post on 11-Oct-2020
When do we need to use QM?
1) Wavelength λ ≈ dimensions of the system
2) Energy level spacing >> kT
( ) /i ji kTi
j j
n g en g
ε ε− −=
populationsDegeneracies (two or more levels with the same energy)
Can treat the system classically if energyspectrum ≈ continuous
kT
quantum classical
Chapter 1
Boltzmann eq.
Classical waves 2 2
2 2 2
( , ) 1 ( , )v
x t x tx t
∂ Ψ ∂ Ψ=
∂ ∂
Ψ(x,t) = A sin(kx – ωt)(show that this satisfies the wave eq.)
2k πλ
= = wave vector
ω = 2πυ = angular freq.
Add two travelling waves of same freq. and amplitude, opposite direction
[sin( ) sin( )]
2 sin cos ( ) cos
A kx t kx t
A kx t x t
ω ω
ω ψ ω
Ψ = − + +
= =
standing wave (fixed nodes)Complex representation
( ')i kx tAe ω φ− +Ψ = Euler: cos sinie iα α α= +
(wave equation) ν = velocity
1i = −
2 2
2 2 2
1vx t
∂ Ψ ∂ Ψ=
∂ ∂
2 2
2 2
( ) ( ) 0v
d x xdxψ ω ψ+ =
( , ) ( ) cosx t x tψ ωΨ =for classical standing wave
2 2
2 2
2 2 2
2 2
2 2
2
4 0
4 0
2
ddx
d pdx h
d V Em dx
ψ π ψλ
ψ π ψ
ψ ψ ψ
+ =
+ =
− + =
hp
λ =
Derivation of the Schrödinger eq.
2hπ
=
2
( )2p V x Em+ =
time independent S. E.
Substitute:
Substitute
ν = v λ
Substitute:
Classical expression for total energy
and
time-dependent S. E.
i Et
∂Ψ= Ψ
∂ Form of wavefunction for a stationarystate
Energy is constant over time.Ψ is a soln. of the time-indep. SE
In QM, all observables are associated with operators
ˆ n n nO aψ ψ=
operator eigenvalueeigenfunction
In QM the eigenvalues correspond to the observables and are real
Eigenvalue eq.
/ ( , ) ( ) iEtx t x eψ −Ψ =
z= x +iy, i =√(-1)
Example of a complex number
2 2
22d V E
m dxψ ψ ψ−+ =
H Eψ ψ= H is the Hamiltonian operator
is an e.f. of ?ikx ikx dAe Bedx
ψ −= +
const. ikx ikxd ikAe ikBedxψ ψ−= − ≠ No
is it an e.f. of 2
2 ?ddx
22 2 2
2ikx ikx ikx ikxd k Ae k Be k Ae Be
dxψ − −⎡ ⎤= − − = − +⎣ ⎦
Yes
vector spacex • y = 0x • z = 0y • z = 0
where x, y, z are vectors in the x, y, z directions
function space*( ) ( )i j ijx x dxψ ψ δ
∞
−∞=∫
1 i = j 0 i j⇒
≠ ≠
The different eigenfunctions of a QM operator are orthogonal(degenerate eigenfunctions are a special case)
If , the functions are normalized* 1i idxψ ψ∞
−∞=∫
Normalize a(a – x) on 0 < x < a
( )2 2 2 2 2 2 2
0 0
3 3 2 52 2 2 2 2 2
0
let ( ) : ( ) 2
3 3 3
a a
a
Na a x N a a x dx N a a ax x dx
x a N aN a a x ax N a
ψ = − − = − +
⎡ ⎤= − + = =⎢ ⎥
⎣ ⎦
∫ ∫
Kroneckerdeltafunction
Orthogonality
52
5
3set 1 3aN N
a= ⇒ =
5
3 ( )a a xa
ψ = − is normalized on 0 < x < a
Orthonormal set of functions: orthogonal and normalized
1( ) ( )n n
nf x b xψ
∞
=
=∑
( ) ( )n nb f x x dxψ∞
−∞= ∫
11. ( ) ( ) ( ) ( )m m n n
nf x x x b xψ ψ ψ
∞
=
= ∑2. Integrate over both sides
The EF’s of a QM operator form a complete set⇒ any function in that space can be written in
terms of the eigenfunctions
bn is the projection of f onto
The analogue in vector spaces is: v = ai +bj +ck
where i, j, k are unit vectors in the x, y, z directions
ψ
nψ
Fourier series
1( ) cos sin2 o n n
n n
n x n xf x b b aL Lπ π
= + +∑ ∑
for a function periodic over –L < x < L
Key ideas:
• time independent and time dependent Schrödinger equations
• operators
• eigenvalue equations
• orthogonal functions and complete basis sets
20 0 0 0 0 0 0 0( , ) ( , )* ( , ) ( , )P x t x t x t dx x t dx= Ψ Ψ = Ψ
probability of finding the particle within dx of x0 at time t0
2( , ) 1x t dx∞
−∞Ψ =∫ probability of finding
the particle somewhere
⇒ Ψ is single valued
Ψ cannot be ∞ over a finite interval
and are continuousddxΨ
Ψ
1. State of QM system completely specified by wavefunction Ψ(x,t)
position:
momentum:
KE:
PE:
total E:
angularmomentum:
22 2
2
2 2
2
2
. . . .
ˆ
ˆ
ˆ2
( )2
ˆ
ˆ ( )
kin
x
pot
mp
x x
pi x
Em x
H V xm x
l y zi z y
E V x
=
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
=
∂=∂
∂= −∂
∂= − +∂
∂ ∂= −∂ ∂
=
ˆ
2. Each observable is associated with a QM operator
All operators for QM observables are Hermitian
As discussed below, Hermitian operators have real eigenvalues
Dirac bra-ket nomenclature
*ˆ ˆm nm n dψ ψ τΩ = Ω∫
Definition of a Hermitian operator
{ }** *ˆ ˆm n n md dψ ψ τ ψ ψ τΩ = Ω∫ ∫
or*ˆ ˆm n n mΩ = Ω
Quantum mechanical observables correspond to Hermitian operators
Eigenvalues of Hermitian operators are real
⇒ complex conjugatei.e., i → -i
*
Show is Hermetiani x∂∂
** * m n n mdx dx
i x i xψ ψ ψ ψ∂ ⎡ ∂ ⎤⎛ ⎞= ⎜ ⎟⎢ ⎥∂ ∂⎝ ⎠⎣ ⎦∫ ∫
|udv uv vdu= −∫ ∫* *
* *|
m n m n
m n n m
dx dxi x i x
dxi x
ψ ψ ψ ψ
ψ ψ ψ ψ∞−∞
∂ ∂=
∂ ∂∂⎡ ⎤= −⎢ ⎥∂⎣ ⎦
∫ ∫
∫
But*
* *
*
n m n mdx dxx x
i i
ψ ψ ψ ψ∂ ∂⎡ ⎤ =⎢ ⎥∂ ∂⎣ ⎦
⎛ ⎞ = −⎜ ⎟⎝ ⎠
∫ ∫
which completes the proof.
*n mdx
i xψ ψ∂−
∂∫
?
(Integration by parts)
A well-behaved wavefunction0 as x → ± ∞
3. In a single measurement of an observable associated withÂ, only an eigenvalue of  can be measured.
4. Expectation value:*
*
ˆˆ A dxA
dx
∞
−∞∞
−∞
Ψ Ψ< >=
Ψ Ψ
∫∫
Average of the observable A, if many measurements are done
If Ψ is an eigenfunction of Â, all measurementsgive the same result
If Ψ is not an eigenfunction of Â
n nb φΨ =∑eigenfunctions of  with eigenvalues an
assuming Ψ is normalized2ˆ ,m mA b a< >= ∑
For normalization
In this case, different measurements give different results
Suppose where are eigenfunctions of Â1 2 1 21 3( ) ( ) ( ), , 2 2
x x xψ φ φ φ φ= +
1 1 1 2 2 2ˆ ˆ, A a A aφ φ φ φ= =
How frequently do we measure a1? a2?
5. The time evolution of a QM system is given by
( , ) ( , )i x t H x tt
∂Ψ= Ψ
∂ˆ
If is a solution of the time-independent SE
/( ) iEtx eψ −Ψ =
ψ
dtdixV
dxd
mH Ψ
=Ψ+Ψ
−=Ψ )(2
ˆ2
22
Can we separate x, t ( , ) ( ) ( )?x t x tψ θΨ =
2 2
22d dV i
m dx dtθ ψ θθψ ψ−
+ =now multiply by
1θψ
dtdixV
dxd
mθ
θψ
ψ1)(1
2 2
22
=+−
• Both sides must be equal to a constant, E• Separates into two ordinary differential equations:
(1)
which may be written in this form
(2)
)()()()(2 2
22
xExxVdx
xdm
ψψψ=+−
dtdixV
dxd
mθ
θψ
ψ1)(1
2 2
22
=+−
θθ Edtdi =
Left-hand side is independent of t and right-hand sideis independent of x
/( , ) ( ) iEtx t x eψ −Ψ =
Simultaneous Observables
The values of two different observables, A and B, can be simultaneouslydetermined (precisely) only if the measurement does not change the stateof the system.
ˆ ˆA a b B↔ ↔
( )
ˆ ˆˆ ˆ( ) ( ), if an e.f. of ( )
ˆ , if also an e.f. of
n n b n n n n
n n n n n n n
BA x B x A A
B B
ψ α ψ ψ ψ α ψ
β α ψ ψ ψ β ψ
= =
= =
⇒ ˆ ˆˆ ˆn nBA ABψ ψ=
ˆ ˆ ˆˆ ˆ ˆ( ) [ , ] 0AB BA A B− = =
commutator
⇒ Two operatorscommute
simultaneousobservables
px, x cannot be known exactly
px, H cannot be known exactly unless V = constant)
Uncertainty principle (Heisenberg)
2p xΔ Δ ≥i ≠ 0 because and do not commuteˆ xp x
Δp and Δx can be associated with standard deviations
2 22 2
22 2
1
1 ( )N
ii
x x x p p p
x x x x xN =
Δ = − Δ = −
Δ = − = −∑
spread in x
x x
x x
p p
δ
δ
= Δ
= Δ
In general
If two operators obey
[A, B] = iC
1A B2
C⇒Δ Δ ≥
[ ],d i HdtΩ= Ω
If [H, Ω] = 0, Ω is called a constant of the motion
If A, B commute, C = 0
If Ω does not depend on time, the time evolution of its average is given by
1s
s s>< =∑ Completeness relation (closure relation)
sr AB c r A s s B c=∑
| |
| | |
n nn n
n nn
mn n mn
H E
H c n E c n
c m H n E c m n
H c E c
ψ ψ> = >
> = >
=
=
∑ ∑
∑ ∑
∑
Here we expand the wave function in the basis set n⟩
1 1 2 2
11 12 1 1
21 22 2 2
1 2
... - - - -
- - - - - -
m m mn n m
n n n n
H c H c H c EcH H c cH H c c
E
H H c c
+ + + =
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟=⎜ ⎟⎜ ⎟ ⎜ ⎟⋅ ⋅⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
Can we find functions that diagonalize the Hamiltonian matrix
11 12 1 1
21 22 2 2
11 12 1
21 22 2
0
H H c cE
H H c c
H E H cH H E c
⎛ ⎞⎛ ⎞ ⎛ ⎞=⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠
−⎛ ⎞⎛ ⎞=⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠
Diagonalizing a 2 x 2 Hamiltonian
⇒ c1, c2, etc. = 0, trivial solutionor
( )( )
( )
11 12
21 22
211 22 12
2 211 22 11 22 12
0
0
0
H E HH H E
H E H E H
E E H H H H H
−= ⇒
−
− − − =
− + + − =
AssumingH12 = H21
( )2 211 2211 22 12
1 42 2
H HE H H H±
+= ± − +
Plug E+ into the system of equations →
Plug E- into the system of equations →
1 2,c c+ +
1 2,c c− −
Complex numbers• |z| = distance from z to
0 in the complex plane:
• For a real number x, |x| = distance from x to 0 on the real number line.
22 yxz +=
( )( )2 2 2
2 2 2
*z z z x iy x iy x y
z z x y
= = + − = +
= = +