Chapter 1

Foundations

1.1 Heisenberg-Born-Jordan Matrix Mechan-ics (1925)

Problem:Find possible energy levels, corresponding stationary states and transitionprobabilities.Idea:Starting from the classical hamiltonian function h(q1, · · · , qk, p1, · · · , pk) ofk generalised coordinates and their conjugated momenta we seek a systemof 2k hermitian matrices {q(1), · · · , q(k), p(1), · · · , p(k)} and a Hamiltonianmatrix H which is diagonal and the diagonal entries of H are the possibleallowed energy levels of the system. In addition the entries q(i)nm, p

(i)nm of the

matrices {q(i), p(i)} determine, in a certain way, the transition probabilities(or rather amplitudes) from the state with energy Em to a state with energyEn < Em. For example, if we consider the two-level system

H =

✓

E2 00 E1

◆

and q =

✓

0 r21r⇤21 0

◆

(1.1)

then the Heisenberg equation

�i~ d

dtq = [H, q] (1.2)

1

2 CHAPTER 1. FOUNDATIONS

perdicts an oscillation between the two energy levels

r12(t) = r12(0)ei!21t (1.3)

with frequency Em�En =: ~!nm in qualitative agreement with atomic spec-troscopy. On the other hand, applying (1.2) to a single particle with mass mand Hamiltonian

h(q, p) =p2

2m+ V (q) (1.4)

consistency with the canonical equation of motion mq = p then requires thecommutation relations

[q(n), p(m)] = i~�nm1 (1.5)

with all other commutators vanishing (i.e. the Heisenberg Algebra).Note that in Heisenberg’s matrix mechanics no mention is made of a

Hilbert space. The essence here is the recognition that in order to bringthe dynamics of the interaction of the atom with the radiation field theintroduction of matrix-valued coordinates is required .

1.2 Dirac Quantization Rules

The transition from classical mechanics to quantum mechanics is most clearlysummarized with the help of Dirac’s quantization rules (1926):

~q ! ~q

~p ! ~p

~q = {~q,H} ! ~q =i

~ [H, ~q]

~p = {~p,H} ! ~p =i

~ [H, ~p] (1.6)

More generally one can consider time independent, matrix valued functionof A(~q, ~p). In this case there is some ambiguity in the quantum mechanicaloperator A since it is uniquely defined by the classical function, A(~q, ~p) on thephase space only once a certain prescription for the ordering of the operatorshas been fixed. Whatever ordering we choose (1.5) and (1.2) imply

�i~ d

dtA = [H, A] (1.7)

1.2. DIRAC QUANTIZATION RULES 3

with solution

A(t) = U †(t)A(0)U(t) (1.8)

where

U(t) = e�i~Ht (1.9)

is the evolution operator.Since matrices are linear operators on some vector space V , we may inter-

pret {q, p, H} as hermitian operators on some vector space V with a completeset of eigenvectors1 commonly denoted by (|~x >, |~k >, |n > )

qi(t)|~x > = xi(t)|~x >

pi(t)|~k > = ki(t)|~k >

H|n > = En|n > (1.10)

If we furthermore endowed V with a positive definite scalar product, < ,>,then, for a generic linear superposition |↵ > of eigenvectors we have for theexpected position and the expected momentum (Born)

~q↵(t) =< ↵|~q(t)|↵ >

< ↵|↵ >

~p↵(t) =< ↵|~p(t)|↵ >

< ↵|↵ >(1.11)

which, in turn, gives the equivalence with Heisenberg’s matrix mechanicsAlternatively, using the completeness of the (delta-function) normalized

eigenvectors of q (for the sake of notational simplicity we consider the caseof one degree of freedom, i.e. a particle in one space dimension)

1 =

Z

dz|z >< z| (1.12)

where |z >< z| is the projector on the Eigenstate |z > to write

< ↵|q(t)|↵ > =

Z

dz

Z

dz0 < ↵|z0 >< z0|q|z >< z|↵ >

=

Z

dz z(t) | < z|↵ > |2 (1.13)

1We give a more mathematical treatment in chapter 2. For now we ignore issues aboutseparability, closure etc.

4 CHAPTER 1. FOUNDATIONS

which leads us to interpret the absolute square of thewave function ↵(z) ⌘<z|↵ > as the probability to find the particle at the point z during a mea-surement process.

1.3 Schrodinger Equation

By means of the evolution operator (1.9) the expectation value (1.11) can beinterpreted in two equivalent ways

< ↵|~q|↵ > (t) =

(

< ↵|~q(t)|↵ > Heisenberg picture

< ↵(t)|~q|↵(t) > Schrodinger picture(1.14)

where

↵(t) >= U(t)|↵(0) > (1.15)

The Schrodinger wave function

↵t(~q) ⌘ ↵(t, ~q) =< q(0)|↵(t) >then satisfies the Schrodinger equation (Homework problem)

i~ @@t↵(t, ~q) = H(

~ir~q, ~q) ↵(t, ~q) (1.16)

This is not, however, the way Schrodinger obtained his wave equation.Instead he obtained it through a clever re-interpretation of the Hamilton-Jacobi-equation in analogy to the relation between geometrical optics andwave optics: For a point particle moving in a potential V (~q) we vary theon-shell action w.r.t. to the end point. Using

pi(t) =@S(qi, t)

@qi(t)|e.m. (1.17)

as well as

dS

dt=@S

@t+X

i

@S(qi, t)

@qi(t)qi(t) =

@S

@t+X

i

piqi (1.18)

and

dS

dt= L(qi, qi, t) (1.19)

1.3. SCHRODINGER EQUATION 5

the on-shell action S(qi, t) satisfies the time dependent Hamilton-Jacobiequation

@S

@t= �h(qi,

@S(qi, t)

@qi, t) (1.20)

If the hamiltonian is time-independent this then leads to the time-independentHamilton-Jacobi equation along the classical trajectory

h(qi,@S0(qi)

@qi) = E (1.21)

where S(qi, t) = S0(qi) � Et and E = const along the classical trajectory.For

h =~p2

2m+ V (~q) (1.22)

this equation is then solved by

(rS0)2 = 2m(E � V ) (1.23)

Geometric Interpretation:

• Since

~q =rS0

m(1.24)

the particle moves along trajectories orthogonal to the surfaces S0 =constant.

• Let r be the geodesic length of the trajectory, then we have

dS0

dr= |rS0| =

p

2m(E � V ) (1.25)

Therefore, the propagation of S = constant has the phase velocity

c(~q) =dr

dt=

Ep

2m(E � V )(1.26)

6 CHAPTER 1. FOUNDATIONS

Schrodinger interpreted these equations in analogy with the transition be-tween wave optics and geometric optics: Let (t, ~q) be a harmonic solutionof the scalar wave equation

1

c2@2

@t2 = � (1.27)

The Ansatz

(t, ~q) = e�i!t�(~q) (1.28)

together with k2(~q) ⌘ !2

c2(~q) then leads to the time-independent equation

(�+ k2)�(~q) = 0 (1.29)

Writing furthermore

�(~q) = A(~q)eiS0(~q) (1.30)

this equation is equivalent to the two real equations

�A� (rS0)2A+ k2A = A�S0 + 2rA ·rS0 = 0 (1.31)

Geometric optics is applicable provided |�AA| << k2. In this limit the phase

S0 the satisfies the eikonal equation

(rS0)2 = k2(~q) (1.32)

In geometric optics we thus have a bundle of light rays whose trajectoriesare orthogonal to the surfaces of constant phase and S = S0�!t = constantevolves with phase velocity c. Schrodinger interpreted in (1926) a materialpoint as a ”material ray” in a medium with phase velocity c(~q) for a time-independent wave equation

(�+2m(E � V )

~2 )�(~q) = 0 (1.33)

i.e. the Schrodinger equation. For A = const the Schrodinger equationreproduces the Hamilton-Jacobi equation. More generally the ”material ray”(in analogy with geometric optics) is a good approximation provided variationof the potential are small over many (de Broglie) wave lengths.

1.4. WKB-APPROXIMATION 7

1.4 WKB-approximation

Schrodinger’s acrobatics between classical and quantum mechanics leads nat-urally to the WKB-approximation. With the Ansatz

(~q) = exp[i

~S0(~q)] , S0(~q) =1X

n=0

✓

~i

◆n

S(n)0 (~q) (1.34)

for the time-independent Schrodinger wave function one shows (Homework

problem) that S(0)0 (~q) solves the time-indep. H-J equation (note that �A

now has an extra factor of ~2 in (1.31)) and with ⇢(~q) = e2S(1)0 (~q) one finds

that ⇢ satisfies the continuity equation

@⇢

@t+r ·~j = 0 , ~j = ⇢~v (1.35)

In 1-dimension this equation solved by

⇢(q) =const.

p

E � V (q)(1.36)

in semi-classical approximation the wave function in 1-dimension is thusgiven by

(q, t) =const.

(E � V (q))1/4e± i

~R

�cl

dr0p

2m(E�V (q(r0)))� iEt~

(1.37)

In the the classically forbidden region one verifies easily that

(q, t) =const.

(V (q)� E)1/4e± 1

~R

�cl

dr0p

2m(V (q(r0))�E)� iEt~

(1.38)

provides an approximate solution to the Schrodinger equation. This solutionapplies, for example, in the calculation of the tunneling probability througha potential barrier.

In generic applications, the region between the classically allowed andforbidden regions is often problematic if the the condition

� <<E � V (~q)

|rV | (1.39)

for the wavelength does not hold.

8 CHAPTER 1. FOUNDATIONS

Historical Notes

The equivalence between Heisenberg’s matrix mechanics, inspired by the dis-creteness of the energy levels of N.Bohr’s ”old quantum theory” and andSchrodinger wave mechanics akin of de Broglie’s particle-wave dualitywas not immediately clear but was established in a series of papers by Bornand Jordan. The probability interpretation of | (~q)|2 was first suggested byBorn.

Although Dirac was widely admired as a scientific magician, many physicist-especially in Berlin and Gottingen - found his language impenetrable, hisreason hard to fathom and his manner cold and distant. Albert Einsteinwas among those who where perplexed: ”I have trouble with Dirac. Thisbalancing between genius and madness is awful”. Niels Bohr, impressed byDirac though puzzled by his indi↵erence to philosophical questions about thenew theory said ”he was the strangest man who ever visited my institute”.

Further Reading

A good book on the foundations, historical development and mathematicalconcepts, although written in somewhat old fashioned language from a mod-ern point of view is J.v. Neumann, Mathematical Foundations of QuantumMechanics. A detailed description of the WKB-approximation is given in L.Schi↵’s book on Quantum Mechanics. It can be used for instance to calcu-late the exciation levels of quarkonium (see Sakurai, ”Modern QuantumMechanics”). More details about the perturbative solution of (1.31) can befound in Courant-Hilbert, ”Methods of Mathematical Physics I”.

Chapter 2

Black-Body Radiation

2.1 Planck distribution (1901)

It was known already before 1900 that the radiation field in a black bodycan be described in terms of independent oscillators for each wave vector andpolarization with number density

n(⌫) =8⇡⌫2

c3(2.1)

To see this we work in Coulomb gauge,

r · ~A(~q, t) = 0 (2.2)

In the absence of sources, the vector potential solves the wave equation⇤ ~A(~q, t) = 0 and is related to the electro-magnetic field through

~E = �1

c

@ ~A

@t, ~B = r^ ~A (2.3)

We take the cavity to be a cube of length L and assume periodic boundaryconditions for the electromagnetic field, ~A(~q + ~nL, t) = ~A(~q, t) for ~n 2 Z3.The general solution to the wave equation is the given by (classical ED, eg.Jackson)

~A(~q, t) =1pL3

X

~k= 2⇡L ~n

2X

�~k=1

~e(~k,�~k)⇣

a(~k,�~k)e�i(!(~k)t�~k·~q) + a⇤(~k,�~k)e

i(!(~k)t�~k·~q)⌘

9

10 CHAPTER 2. BLACK-BODY RADIATION

(2.4)

with !(~k) = c|~k| and real polarisation vectors ~e(~k,�~k), orthogonal to~k and

~e(~k,�) · ~e(~k,�0) = ���0 (2.5)

A straight forward calculation shows that (in Gaussian units) the classicalenergy of the electromagnetic field is then given by

h( ~E, ~B) =1

8⇡

Z

( ~E2 + ~B2)d3q =1

2⇡

X

~k

2X

�~k=1

~k2|a(~k,�~k)|2 (2.6)

Let us now define the real variables

q(~k,�~k, t) =1

cp4⇡

[a(~k,�~k)e�i(!(~k)t + a⇤(~k,�~k)e

i(!(~k)t]

p(~k,�~k, t) =�i

cp4⇡

[a(~k,�~k)e�i(!(~k)t � a⇤(~k,�~k)e

i!(~k)t] (2.7)

The classical energy of the electro magnetic field (2.6) then becomes

h =1

2

X

~k,�~k

n

p2(~k,�~k, t) + !(~k)2q2(~k,�~k, t)o

(2.8)

i.e. that of a countable infinity of independent classical harmonic oscillators.Furthermore, the canonical equations

q(~k,�~k, t) =@h

@p(~k,�~k, t)

p(~k,�~k, t) = � @h

@q(~k,�~k, t)(2.9)

are equivalent to the Maxwell equations in Coulomb gauge.By assigning an average energy kBT to each oscillator Lord Rayleigh and

J.H. Jeans wrote an expression for the energy in thermal equilibrium with areservoir of temperature T ,

eclass(⌫, T ) = n(⌫)kBT (2.10)

2.2. QUANTUM MECHANICAL DESCRIPTION 11

which was in satisfactory agreement with observations for small frequencybut leads to a diverent total energy density

Z

d⌫eclass(⌫, T ) = 1 (2.11)

Planck justified his distribution formula

e(⌫, T ) = n(⌫)h⌫

eh⌫

kBT � 1(2.12)

with the assumption that the possible energy levels of a harmonic oscillatorwith frequency ⌫ are discrete

✏n = nh⌫ (2.13)

Including the Boltzmann-factor

wn =e� ✏n

kBT

1P

n=0e� ✏n

kBT

(2.14)

for the probability of the n-th excitation we obtain the average energy

e(⌫, T ) = n(⌫)1X

n=0

✏nwn = n(⌫)h⌫

eh⌫

kBT � 1(2.15)

instead of (2.10).

2.2 Quantum mechanical description

In the quantum mechanical description we associate a radiation field withfixed wave vector (and polarization) energy ✏n to n photons and this stateis described by a vector |n > 2 V . The vector space V is spanned by theinfinite dimensional complex linear combinations

| >=1X

n=0

n|n > (2.16)

12 CHAPTER 2. BLACK-BODY RADIATION

with a positive definite inner product

< |� >=1X

n=0

⇤n�n =< �| >⇤ (2.17)

This inner product defines a norm

|| | > || =< | > (2.18)

and a distance || | > �|� > || and therefore a (strong) topology. Thequantum mechanical Hilbert space, H is then obtained by completing thespace V , including all convergent Cauchy series, | n > 2 V with

|| | n > �| m > || ! 0, m, n ! 1 (2.19)

Completeness can be characterized by the following equivalent condition: if

a series of vectors converges absolutely,1P

n=0|| n|n > || < 1, then the series

converges in H, in the sense that the partial sums converge to an element of

H. Each vector in H has then a unique expansion | >=1P

n=0 n|n > with

1P

n=0| n|2 < 1.

2.3 Pure states, linear operators

A pure state | > satisfies || | > || =1P

n=0| n|2 < 1 = 1 and | n|2 is the

probability to find n photons in | > during a measurement. A pure stateis thus an equivalence class of vectors

{ei↵| > | ↵ 2 R , || | > || = 1} (2.20)

In mathematical terms, quantum mechanical states form projective repre-sentations.

Linear operators on H are linear maps

A : H ! H (2.21)

2.3. PURE STATES, LINEAR OPERATORS 13

with a domain D(A). In quantum mechanics we often encounter un-bounded operators, i.e. with infinite norm

sup| >2H|| A| > |||| | > || = 1 (2.22)

as, for example, for the number operator N , N |n >= n|n >. A naturaldomain for N is

D(N) = {| >2 H |1X

n=0

n2| n|2 < 1} (2.23)

The adjoint operator to A is defined through

< A† |� >=< |A|� > (2.24)

for all |� >, | > 2 D(A). It is unique, provided D(A) is dense in H. Aself adjoint operator satisfies A† = A.

If an operator A has a complete set of eigenvectors with real eigen-values then A is self adjoint. There are, however more general, infinitedimensional operators, eg. the position operator, q, whose eigenvectors arenot in H. More generally, the spectrum of a bounded operator is a gener-alisation of the concept of eigenvalues for matrices. Specifically, a complexnumber c is said to be in the spectrum of a bounded linear operator A ifcI � A is not invertible. The hamilton operator for black body radiationwith frequency ⌫

H = h⌫N = ~!N =1X

n=0

~!|n >< n| (2.25)

is self adjoint.The trace of an operator is defined analogously to the finite-dimensional

case (if it exists) by

Tr(A) =1X

n=0

< n|A| n > (2.26)

Important note: The most general state of a quantum mechanical sys-tem is not a pure state but is instead described by a density matrix which,

14 CHAPTER 2. BLACK-BODY RADIATION

in a suitable basis, can be written as

⇢ =1X

n=0

wn| n >< n|, wn � 0,1X

n=0

wn = 1 (2.27)

so that wn is the probability to find the system in the state | n > during ameasurement with an apparatus that consists of projectors on n (e.g. SternGerlach, see chapter 3). The expectation value of A in ⇢ is given by

< A >⇢= Tr(A⇢) (2.28)

Quantum mechanical states can be mixed: If ⇢1, ⇢2 � 0 and c1, c2 � 0 suchthat c1 + c2 = 1, then

⇢ = c1⇢1 + c2⇢2 (2.29)

is again a possible quantum mechanical state. Every state ⇢ can be writtenas a mixture of pair-wise orthogonal states. ⇢ is a pure state if and only if⇢2 = ⇢.

2.4 Ladder operators

These are the creation - and annihilation operators

a†|n >=pn+ 1|n+ 1 >, a|n >=

pn|n >, a|0 >= 0 (2.30)

with domain

D(a†) = D(a) = {| > |1X

n=0

n| n|2 < 1} (2.31)

The above relations imply, in particular,

a†a = N and [a, a†] = 1 (2.32)

The algebra of a† and a is then isomorphic to the Heisenberg algebra (1.5)in 1-dimension as is shown by the substitution

p = i

r

~!2(a† � a)

q =

r

~2!

(a† + a) (2.33)

2.5. QUANTUMMECHANICAL DESCRIPTION OF BLACK-BODY RADIATION15

with the simple action (see homework problem 1) on the wave functions n(x) ⌘< x|n >. This change of basis establishes furthermore the isomor-phism between the Hilbert-spaces `2 of n-photon modes with fixed frequencyand L2 of square integrable functions. The Hamiltonian now becomes

H =1

2p2 +

!2

2q2 � ~!

2(2.34)

This then establishes the equivalence between the dynamics of n-photonmodes with fixed wave vector, ~k (and polarization) and the 1-dimensionalharmonic oscillator.

2.5 Quantum mechanical description of black-body radiation

Generalising the above to the infinite tensor product of non-interacting pho-ton modes with arbitrary wave vector, ~k (and polarization) Hilbert-space

He.m. = ⌦~k,�~kH~k,�~k

(2.35)

defines a free relativistic quantum field theory. This is done by replacing~A(~q, t) by the generalised operator valued function

~A(~q, t) =X

~k,�~k

~c22⇡!(~k)L3

!

12

~e(~k,�~k)⇣

a(~k,�~k)e�i(!(~k)t�~k·~q) + a†(~k,�~k)e

i(!(~k)t�~k·~q)⌘

(2.36)

with

[a(~k,�~k), a†(~k0,�0~k)] = �~k,~k0��~k,�0~k

1 (2.37)

The normalization chosen in (2.36) ensures that for a plane wave, where allFourier modes propagate in the same direction, the canonical pairs

~A(~q, t), ~P (~q, t) = � 1

2⇡c2˙~A(~q, t) (2.38)

satisfy the equal time commutation relations (field theory Heisenbergalgebra)

[Ai(~q, t), Pj(~q0, t)] = i~�ji �3(~q � ~q0)P~k (2.39)

16 CHAPTER 2. BLACK-BODY RADIATION

where P~k is the projector onto the plane orthogonal to~k

|~k| .Returning to the general case the Hamilton operator becomes after nor-

mal ordering

H =X

~k,�~k

~!(~k)a†(~k,�~k)a(~k,�~k) =X

~k,�~k

~!(~k)N(~k,�) (2.40)

which is that of the sum over independent harmonic oscillators with wavevector ~k, and polarisation �~k.

The operator valued quantum field ~A(~q, t) evolves according to

~A(~q, t) = ei~Ht ~A(~q, 0)e�

i~Ht (2.41)

and consequently the electric and magnetic field operators satisfy the Heisen-berg equation

�i~ d

dt~E(~q, t) = [H, ~E(~q, t)] � i~ d

dt~B(~q, t) = [H, ~B(~q, t)] (2.42)

If we place the cavity in a heat bath of temperature, T, the density matrixfor the electromagnetic field is given by

⇢ =1

Z(T )e� 1

kBT H, Z(T ) = TrHe.m.e

� 1kBT H

(2.43)

The occupation number for the photon mode (~k,�) is determined by thedensity matrix (2.43) so that

< N(~k,�) >⇢=1

e~!(~k)kBT � 1

(2.44)

which leads to the spectral energy density e(!, T ) of the black body radiation(2.12).

2.6 Coherent States

In practical applications an important role, especially in quantum optics, isplayed by the quasi classical or coherent states. They are eigenstates ofthe annihilation operator

a|z >= z|z >, z 2 C (2.45)

2.6. COHERENT STATES 17

and are created by the family of operators

V (z) = eza†

(2.46)

In field theory we may take arbitrary tensor products coherent states |z(~k,�) >.Then with

~E(~q, t) = iX

~k,�

~!(~k)2⇡L3

!

12

~e(~k,�)⇣

a(~k,�)e�i(!(~k)t�~k·~q) � a†(~k,�)ei(!(~k)t�~k·~q)

⌘

(2.47)

the expectation value < z(~k,�)|Ei|z(~k,�) > describes a monochromatic,coherent (quasiclassical) light wave used in laser physics.

Historical Notes

The mathematical concept of a Hilbert space was named after David Hilbert(1862-1943, Gottingen), by John von Neumann.

Although Planck was first to propose the quantization of the energy ofthe electro magnetic field he was reluctant to interpret the field as particles(photons). The particle like nature of the electro-magnetic field was firstproposed by Einstein in 1905 (at that time working at the patent o�ce inBern). This interpretation was, however, disputed universally until 1919 withRobert Millikans detailed experiments on the photoelectric e↵ect (see section5.3), and with the measurement of Compton scattering. Einstein receivedthe Nobel price for his prediction in 1922.

Further Reading

A detailed discussion of the quantization of the electromagnetic field can befound in J.J. Sakurai, ”Advanced Quantum Mechanics”, chapter 2.