Summary of quantum mechanics 1 Classical mechanics

Summary of quantum mechanics

1 Classical mechanics

• Describe the state of a particle by its position x(t) and velocity v(t)=dx/dt= x& as functions of time t

• Hamilton’s equations of motion (equivalent to Newton’s law F =ma)

p q H

∂

=∂

& and

q p H

∂

−∂

=

&

• Hamilton function=total energy

V T p q

H( , )= +

2 Experimental background

Quantum mechanics assumes two relations to apply generally for all particles:

1. Photoelectric effect: light consist of particle-like discrete quantities of energy that obey ω

=h

=hf E

2. Compton effect: photon momentum is

h k p= =h

λ

This relation associates a wavelength λ to any particle, called the de Broglie wavelength.

3. Motivation of Schrödinger equation:

m k dk

v_group d h

=

= ω

Integrate:

m E k

m k

2 2

2 2

2 h

h h

=

=

= ⇒ ω

ω

This relation agrees with the solution ψ(x,t)= Ae^i(kx−ωt) to the Schrödinger equation

i t x

m ∂

= ∂

∂

− ∂ ψ ψ

h h

2 2 2

2

In a potential the Schrödinger equation generalizes to

i t x V

m ∂

= ∂

∂ +

− ∂ ψ ψ ψ

h h

2 2 2

2

4. Group velocity vs phase velocity

Classical wave equation in one dimension

2 2

2 2

2 1

t f c x

f

∂

= ∂

∂

∂

Examples of solutions: f = Acos(kx−ωt), f = Asin(kx−ωt), f = Ae^i(kx−ωt)

provided that ω² =c²k². Solutions with ω =±ck travel in the ±x direction. The velocity of such sinusoidal waves, c=ω/k is called the phase velocity.

A relation beween frequency and wave number is called a dispersion relation. In general, a wave can have a nonlinear dispersion relation, ω(k), and is then called a dispersive wave.

In such cases another definition of wave velocity is used, so called the group velocity:

dk v_{group =} dω

The group velocity is the velocity of a wave packet, i.e., a collection of waves in a narrow k interval that propagate together in constructive interference. To motivate the definition, consider a wave packet of two waves with wave numbers k₁, k₂ and frequencies ω₁,ω₂^{. At} the point x of constructive interference, the waves stay in phase with

k t x k

t x k t x

k 









−

= −

− ⇒

=

−

2 1

2 1 2

2 1 1

ω ω ω

ω

which gives d /ω dk ^if k1−k2 is small.

5. Probability, discrete case

Discrete case

Assume that a discrete random process can produce values x₁,x₂,x₃,...,x_N, and that in a very large number of observations of x, the result x is found _j n times. _j

The probability that an individual measurement of x gives the result x is defined to be the _j fraction P_i =n_i/n, where n⁼∑_i^N₌ ni

1 is the total number of observations.

The probability distribution P has two basic properties:

1. Positive definiteness: P_i ≥0

2. Normalization: ₁ = ¹ +...+ = ∑ ¹ =1

∑₌ ⁼

n n n

n n

P n

N

i i

N N

i i

The average value of all the measured results is given by the expectation value x ⁼∑_i^N₌ xiPi 1

The likely spread in the outcomes around this expectation value is called uncertainty x∆ in x, and defined as the standard deviation whose square is called the variance

( )∆^{x 2} =∑_i^N₌₁(^xi − ^x )^2Pi = ^x2 − ^{x 2, where} x^{2 =}∑_i^N₌₁xi²Pi.

6. Probability, continuous case

Consider a random process that can give any real number x as a result of a measurement.

Divide the range of possible values into finite subintervals, each of length dx . Let P(x)dx be the relative frequency of measured results in dx around x.

In the limit when dx goes to zero, this procedure defines a continuous function P(x) called the probability density, and P(x)dx is the probability to find a value in dx around x.

The probability density is positive definite, P(x)≥0, and normalized, ∫₋^∞_∞^P(^x)^dx=1^. The expectation value of x is given by ^∞∫

∞

−

= xP x dx

x ( )

The uncertainty, x∆ , is the standard deviation, which is the square root of the variance

( )^{∆x 2 =} ∫^{∞(x− x )2P(x)dx= x2 − x 2}

∞

−

, where ^{x2 =}∫₋^∞_∞^x2P(x)dx.

7. Hermitean operators

Definition: An operator Aˆ is called a Hermitean if

∫

∫^{ψ*Aˆϕdx= (Aˆψ)*ϕdx} for all ψ^,ϕ

Theorem: Hermitean operators have the following properties:

(a) The eigenvalues are real

(b) Eigenfunctions belonging to different eigenvalues are orthogonal: ∫^ϕ*ψdx=0

(c) The eigenfunctions form a complete set because it is always possible to express any wave function as a linear superposition of eigenfunctions, which is a generalized Fourier series of the form

=∑

= −

Ψ

,...

3 , 2 , 1

) /

( )

, (

n

t iE n n

e n

x c t

x ψ ^h

where the coefficients c are complex constants. _n Proof of (a),(b):

0 )

( ˆ ) (

ˆ

*

*

*

*

*

*

*

=

−

⇒

=

=

=

=

∫

∫

∫

∫

∫

dx b

a

dx b

dx A

dx a

dx A

ψ ϕ

ψ ϕ ψ

ϕ

ψ ϕ ψ

ϕ

(a) If a=b, then we get ^*

1

*

*) 0

(a−a dx= ⇒a=a

=

∫4243

1ψ ψ , which shows that a is purely real.

(b) If a≠b, then we get ( ) ^* 0 ^* 0

0

=

= ⇒

− ∫ ∫

≠

dx dx

b

a₂₃ ϕψ ϕψ

1 , which shows that ψ^,ϕ ^{are ON.}

(c) This is a generalized version of Fourier’s theorem, which we accept without proof here.

8. Postulates of Quantum Mechanics

= basic principles that summarize experimental facts, but can not be derived from any more basic rules.

Classical rule Quantum postulate

The state of a particle is given by its position )

(t

x and momentum p(t).

The state of a particle is represented by its wave function Ψ( tx, ). Position and momentum are represented by Hermitean operators

i x p x

x ∂

− ∂

=

= , ˆ h ˆ

2. Every dynamical variable is a function of )

(t

x and p(t): A= A(x,p).

Dynamical variables are represented by Hermitean operators obtained by

) ˆ , ˆ ˆ A(x p

A= .

3. A measurement of A(x,p) leaves the state )

( ), (t p t

x unchanged.

If the particle is in the state Ψ, then the result of a measurement of A is one of the eigenvalues of A : Aˆψ_a =aψ_{a with} probability

* 2

)

(^{a ∝} ∫ ^Ψdx

P ψ_a

As a result of the measurement the state of the particle changes to ψ_a^.

4. The state variables x,p change in time according to Hamilton’s equations:

p x H

∂

= ∂

& ,

x p H

∂

−∂

=

&

The state Ψ obeys the time dependent Schrödinger equation

i t x V

m ∂

Ψ

= ∂ Ψ

∂ + Ψ

− h ∂ h

2 2 2

2

where Hˆ = H(xˆ,pˆ) is the Hamiltonian, which is a Hermitean operator, and H is the classical Hamilton function.

9. Born probability interpretation and expectation values

Probability of finding a particle in the state Ψ in the interval dx at position x dx

t x dx

t x

P( , ) = Ψ( , )²

Normalization such that the probability of finding the particle somewhere is one:

1 )

, ( )

,

( = ∫ Ψ ² =

∫ ^∞

∞

−

∞

∞

−

dx t x dx

t x P

Position expectation value

∫

∫ ^∞

∞

−

∞

∞

−

Ψ

=

= xP x t dx x x t dx

x ( , ) ( , )²

=average result from measuring the position many times of a particle in the state Ψ. Expectation value of a variable A

∫∞

∞

−

Ψ Ψ

= x t A x t dx A ^*( , )ˆ ( , )

Momentum expectation value:

∫∞

∞

−

Ψ



 





∂

− ∂ Ψ

= x t dx

i x t x

p ^*( , ) h ( , )

10. Born probability interpretation

Assume that Ψ is a superposition of two ON states: Ψ=^c₁ψ₁+^c₂ψ₂^, ∫^∞

∞

−

= ij j

iψ dx δ

ψ^{* .}

Coefficients obtained by ∫ ∫ ∫ ^∞∫

∞

−

∞

∞

−

∞

∞

−

∞

∞

−

Ψ

= + ⇒

=

Ψdx c _i dx c _i dx c_{i i} dx

i

* 2

* 2 1

* 1

* ψ ψ ψ ψ ψ

ψ Normalization:

2 2 *

2 2 2 1 2

* 2 2 2 1

* 1 2 1

2 ∫ ∫ ∫

∫ ^∞

∞

−

∞

∞

−

∞

∞

−

∞

∞

−

Ψ

=

= + ⇒

= +

=

Ψ dx c ψ ψ dx c ψ ψ dx c c P_i c_i ψ_i dx This is the probability that a particle in the state Ψ=^c₁ψ₁+^c₂ψ₂ is in the state ψ_i^. Similar for superpositions of more states.

11. Definite observables (opposite to uncertain)

If the wavefunction is an eigenstate of the operator Aˆψ_a =aψ_{a, then} A is called definite, because then measurements of A gives the result a, again and again.

Expectation value of A is

{A dx a dx a

A _{a a}

a a a

a

=

=

= ∫ ^∞∫

∞

−

∞

∞

− =

ψ ψ ψ

ψ

ψ

*

* ˆ

Uncertainty in A is zero, hence definite:

ˆ 0 )

( ^{2 2 2 * 2 2 2 2}

2

=

−

=

−

=

−

=

∆ ∫^∞

∞

− =

a a a dx A A

A A

a a

a

a123

ψ

ψ ψ

12. Uncertain variables (opposite to definite)

If the wavefunction Ψis NOT an eigenstate of the operator, then A is called uncertain.

Expand Ψ in a superposition of ON eigenfunctions to A :

i j

j i j i

i i i

c dx c

dx c

ij

=

= Ψ

= Ψ

∑ ∫

∫

∑

=

∞

∞

−

∞

∞

− 14243

δ

ψ ψ ψ

ψ

*

*

Normalization:

= ⇒

=

=

Ψ ∑ ∫ ∑

∫ ^∞

∞

−

∞

∞

−

2 1

2 * 2

i i i

i i

i dx c

c

dx ψ ψ

2 2 *

∞∫

∞

−

Ψ

=

= c dx

P_{i i} ψ_i =probability to measure a _i

If the measurement gives the result a , then the wavefunction changes (collapses) to the _i corresponding eigenfunction Aˆψ_a =aψ_a. Then A becomes definite.

13. Heisenberg uncertainty relation

Position-momentum uncertainty relation:

2

≥ h

∆

∆x p

Example of a minimum uncertainty wave function is the Gaussian wave function:

2

, 2 , 2

0 ) 1

(

2 2 2

2 2

2 / ²

2

h

h

=

∆

∆

=

=

=

=

=

Ψ ⁻

x x

p x

p x

e

x ^x

σ σ

π σ

σ

13. Time evolution: stationary states

Energy eigenstates Hψ_E = Eψ_E Time dependence of product state

h h

h h

/ /

2 2 2

) ( ) , (

) (

1 2

1

) ( ) ( ) , (

iEt E

iEt

E E

E

e x t

x e t T

t T i T E x V

m

t T x t

x

−

−

= Ψ

= ⇒

∂ ⇒

= ∂

 =







 +

∂

− ∂

= ⇒ Ψ

ψ ψ ψ

ψ

Energy eigenstates are stationary states, since the probability density is time independent:

2

2 ( )

) , ( ) ,

(x t x t x

P = Ψ =ψ_E

14. Time evolution: nonstationary states

Expand as superpositions of stationary states.

Example:

h

h /

2 /

1

2

1 ( )

2 ) 1

( 2 ) 1 ,

(x t = _E x e^−iEt + _E x e^−iEt

Ψ ψ ψ

( )

[ ]

[

]

h h h

/ ) cos(

2 ) ( )

2 ( 1

Re 2 ) ( )

2 ( 1

) ( )

2 ( ) 1 , (

2 1 2

2 2 1

/ ) 2 (

2 2 1

/ 2 2

/ 1

2 1

2 1

t E E x

x

e x

x

e x e

x t

x P

E E

t E E i E

E

t iE E

t iE E

− +

+

=

= +

+

=

= +

=

−

−

−

ψ ψ

ψ ψ

ψ ψ

From this we identify the uncertainty in time as the decay time of the initial state:

2 1 2

1 ) / ,

(E −E ∆t h≈π ⇒∆E∆t ≈h ∆E ≈E −E

In accordance with the energy-time uncertainty relation

≈h

∆

∆E t

15. Current and continuity equation

Continuity equation

=0

∂ + ∂

∂

∂ x j t P

Probability density:

)2

, ( ) ,

(x t x t

P = Ψ

Probability current dentity



 





∂ Ψ Ψ∂

∂ − Ψ Ψ ∂

= m x x

j i

*

*

2 h

16 Example square well







∞

∞

= 0 ) (x

V

x L

L x x

<

<

<

<

0 0

kx B kx A x x E

L m

x ( ) sin cos

0 2 ₂

2 2

+

=

= ⇒

∂

− ∂

< ⇒

< ^h ψ ψ ψ

Continuity requirements at endpoints:

2 2 2 2

2 2

2 sin )

(

,...

2 , 1 , 0

) (

0 0

) 0 (



 



= 

=

=

⇒

=

=

= ⇒

=

= ⇒

L n m m

E k

L x n x L

n n kL L

B

n n

π ψ π

π ψ

ψ

h h

17 Example reflection against a potential step



=

0

) 0

(x V

V

x x

>

<

0 0

1. Assume E<V₀. Schrödinger equation

x ikx ikx

Ce x E

x V x m

Be Ae

x x E

x m

ψ κ

ψ ψ ψ

ψ ψ ψ

−

−

=

= ⇒

∂ +

− ∂

>

+

=

= ⇒

∂

− ∂

<

) 2 (

: 0

) 2 (

: 0

2 0 2 2

2 2 2

h h

h h

h

h 2 ( )

2 , 2

2

0 0

2 2 2

2 mE mV E

k m V

m

E k −

=

= + ⇒

−

=

= κ κ

(1)

This determines the penetration depth ^1/κ into the classically forbidden region x>0. The amplitudes B,C can be determined from the incoming amplitude A :

continuity of ψ(x=0)⇒ ^A+^B=^C (2)

continuity of = ⇒

dx x dψ( 0)

C ikB

ikA− =−κ (3)

2. Assume E >V₀ The solution is formally the same but, turns into a plane wave for x>0:

h h

) (

2 )

(

2 ₀ m E V₀

E i V

m −

− = κ =

18 Example Tunneling







= 0 0 ) (x V₀

V

x L

L x x

<

<

<

<

0 0

Assume E<V₀. Schrödinger equation

ikx

x x

ikx ikx

Fe x L x

De Ce

x E

x V L m

x

Be Ae

x x E

x m

=

>

+

=

= ⇒

∂ +

− ∂

<

<

+

=

= ⇒

∂

− ∂

<

−

−

) ( :

) 2 (

: 0

) 2 (

: 0

2 0 2 2

2 2 2

ψ

ψ ψ ψ ψ

ψ ψ ψ

κ

h κ

h

h h

h

h 2 ( )

2 , 2

2

0 0

2 2 2

2 mE mV E

k m V

m

E k −

=

= + ⇒

−

=

= κ κ

Which determines the penetration depth ^1/κ^.

Reflection and transmission probabilities: , , 1

2 2

= +

=

= T R

A T F A R B

Continuity at x=0⇒

D C B

A+ = + ikA−ikB=κC−κD Continuity at x=L⇒

L L

ikL

ikL Be Ce De

Ae + ⁻ = ^−κ + ^κ ikAe^ikL −ikBe^−ikL =−κCe^−κL +κDe^κL These equations can be solved to give R,T .

For a wide barrier such that e^−2κL <<1, D≈0 which gives e L

V E V

T E ₂ ^2κ

0

0 )

(

16 − ₋

≈

This explains the exponential dependence of the tunnelling current in Scanning Tunneling Microscopy (STM).

19 Harmonic oscillator

( )

π ω ψ

ω ω

ψ ψ ψ ω

a m e

a x

n n

E m

k

i t x x m

m

a x n

h h

h h

 =



 



=

= +

=

=

∂

= ∂

∂ +

− ∂

− /2 2 2

/ 1

0

2 1 2

2 2 2

2 2

1 , )

(

,...

2 , 1 , 0 , 2 1 2

2 2

The ground state is a minimum uncertainty wavefunction.

20 Angular momentum

1 sin

, 1 ,..., 1 ,

,...

, 1 , , 0

) , ( )

, ˆ (

) , ( ) 1 ( ) , ˆ (

ˆ ˆ ˆ

2 2

0 0

2 3 2 1

2 2

=

− +

−

−

=

=

= +

=

∇

×

−

=

×

=

∫

∫ ^lm

m l m

l z

m l m

l

Y d d

l l l

l m l

Y m Y

L

Y l

l Y

L

r i p r L

π

π θ θ φ

φ θ φ

θ

φ θ φ

θ

h h h

21 Hydrogen like atoms

Schrödinger and others became convinced that quantum mechanics is correct after his successful explanation of atomic spectra, which did not have any previous explanation.

Hydrogen: one electron with charge −e^{and mass} m bound to one proton with charge e:

1 )

) ( (

) 2 (

) 1 ( ) ( 2

) , ( ) ( ) , , (

4 2

0 2

0 2 2

2 2

2 2 2

0 2 2

2

=

=

=

 =



 



 + +

+

−

=

=

−

∇

−

∫

∫ ^∞

∞

dr u dr R r

r r r u R

ER R r mr V

l l dr

rR d mr

Y r R r

r E e m

m l

h h

h

φ θ φ

θ ψ

ψ πε ψ

ψ

6 . ) 13

4 ( ,..., 2

3 , 2 , 1

, _{2 2}

0 4

2 = = =

−

= πε h

m E e

n n

E_n E^R _R eV

e m a

a e r a

u_{n l} ^{r a}

2

2 0 0

/

0 0 ,

0

4

2 ₀

πε h

=







=  ⁻

=

Hydrogen like atom: one electron with charge −e bound to one proton with charge Ze , performing relative motion with effective mass

M m

mM

= +

µ ^:

ψ πε ψ

µ ψ ^{r E}

Ze =

−

∇

−

0 2 2

2

4 2

h

,...

3 , 2 , 1 ) ,

4 (

2 ₀ ^{2 2 2}

4

2 =

−

= n

n e E_n Z

πε µh

µ πε ²

2 0 0

/

0 0 ,

0

4

2 ₀

h a Ze

a e r a

u_{n l} ^{r a}

=







=  ⁻

=

Comment: The Bohr model mainly has historical interest. It is wrong since electrons do not have orbits, which is a classical concept. Use the Schrödinger equation instead.