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

= = + − = +

= = +