Theory and Application of Nanomaterialsahrenkiel.sdsmt.edu/courses/Fall2019/NANO702/... · “state-space”, which allow a description of transitions between quantum states. Figure:

Outline Introduction to Hilbert Space Expectation Values Quantum Harmonic Oscillator Fermi’s Golden Rule Appendix

Theory and Application of NanomaterialsLecture 7: Quantum Primer III, Evolution of the Wavefunction)

S. Smith

SDSMT, Nano SE

FA17: 8/25-12/8/17

S. Smith (SDSMT, Nano SE) Theory and Application of Nanomaterials FA17: 8/25-12/8/17 1 / 25


Introduction

Heisenberg’s matrix formulation of quantum mechanics builds on theconcept of vector spaces, which is a geometrical interpretation of theorthonormal set of functions which satisfy the time-independentSchrodinger equation, allowing an intuitive description of the timeevolution of quantum systems.

Figure: ”The problems of language here are really serious. We wish to speak ofthe nature of the atoms. But we cannot speak about atoms in ordinarylanguage.” – W. Heisenberg.



Outline

1 Introduction to Hilbert Space

2 Expectation ValuesCommutation RelationsSuperposition of StatesTime-dependent expectation values

3 Quantum Harmonic OscillatorAnalytical SolutionAlgebraic SolutionBra-ket notation

4 Fermi’s Golden RuleFirst Order Perturbation TheorySinusoidal PerturbationFermi’s Golden Rule

5 Appendix



Introduction to Hilbert Space

We will now open the discussion to the possibility that the wavefunction may evolve in time. Todo so, we need to introduce the important concepts of superposition of states and“state-space”, which allow a description of transitions between quantum states.

Figure: The wavefunction can be thought of as a state-vector in an n-dimensional space,composed of a super-position of Eigenfunctions 'n(r, t).



Commuting variables

We assume the wavefunction may be in a superpostion of the Eignestates found by solving thetime-independent Schrodinger equation, and that the system evolves ”adiabatically” in time.With such a description, let us calculate the time rate of change of some observablecorresponding to the arbitary operator bA :

dh bAidt

=d

dt

Z

⇤bA d3r =

Z

@

@t

⇣

⇤bA

⌘

d3r =

Z

✓

⇤ @

@t

⇣

bA ⌘

+@ ⇤

@tbA

◆

d3r

=

Z

(

⇤

@ bA

@t + bA

@

@t

!

+@ ⇤

@tbA

)

d3r = h@ bA

@ti+

i

~

Z

h

⇤bH bA � ⇤

bA bH i

d3r

dh bAidt

= h@ bA

@ti+

i

~[ bH, bA]

(1)

The final form shows us that h bAi evolves in time according to the expectation value of thederivative of the operator itself, if it depends on time, plus i/~ times the quantity in brackets,which we call the commutator, defined as:

[ bA, bB] = h bA bBi � h bB bAi

The interpretation of the above is that if the operator bA does not change the system, then theorder of operations is irrelevant and [ bA, bB] = 0 , then we say bA ’commutes’ with the bB.



Quantum Limits on Observables

The interpretation of the above is that if the operator bA does not change the system, then theorder of operations is irrelevant. For example, if [ bH, bA] is equal to zero, then h bAi is a constant.

In that case, we say bA ’commutes’ with the Hamiltonian bH. This is in general not the case. Forinstance, for the so-called conjugate variables {x , p}, which do not commute, one measurementfollowed by the other will not yield the same result as the time-reversed sequence. In fact, thecommutator can be used to derive the uncertainty relations:

�p�x � ~/2

�E�t � ~/2

Evidently, non-commutation of variables tells us that complete knowledge of quantum systems isnot possible, even in principle. This is an important principle of quantum mechanics without aclassical counterpart.



Superposition of States

Admitting the wavefunction may not be stationary, i.e. it may evolve in time, we know it mustevolve between Eigenstates, as the completeness of the Eigenfunctions (see appendix) tells us

that all solutions of the Schrodinger equation bH (r) = E (r) must be decomposable into asubset of these functions { n(r)} :

(r, t) =X

n

cn'n(r, t) where 'n(r, t) = e�i En~ t n(r) (2)

Figure: The wavefunction can be thought of as a state-vector in an n-dimensional space,composed of a super-position of Eigenfunctions 'n(r, t).



Time-dependent expectation Values

It is equally possible to write this superposition by considering the coe�cients cn to be functionsof time cn(t):

(r, t) =X

n

cn(t) n(r) where

bA n(r) = an n(r)

we can then write out the expectation values for the operator bA(t):

h bA(t)i =X

n

an|cn(t)|2 where cn(t) =

Z

⇤n (r) (r, t)d

3r

the coe�cient cn(t) can be thought of as the time-dependent projection of the wavefunction (r, t) onto the stationary state n(r). We can show that this is equivalent to (2) by inserting

the above form for cn(t) into the definition for h bAi:

h bA(t)i =X

n

an

Z

⇤(r0, t) n(r0)d3r 0

| {z }

cn(t)

Z

(r, t) ⇤n (r)d

3r

| {z }

c⇤n (t)

h bA(t)i =Z

⇤(r0, t) bAd3r 0Z

(r, t)X

n

⇤n (r) n(r0)

| {z }

�(r�r0)

d3r =

Z

⇤(r0, t) bA (r0, t)d3r 0

(see appendix for derivation of �(r� r0) term).



Quantum Harmonic Oscillator

We will take one more detour into quantum mechanics. Square potentials can be treatedpiece-wise, the Coulomb potential yields closed form solutions expressed as products of specialfunctions, other more general potentials may be expressed as a Taylor series:

V (x) = Vo + V1(x � xo) +V2

2(x � xo)

2 + ...

The first term in this expansion which may have a bound-state is the quadratic term, and thishas the form of the potential for a mass bound by a spring, i.e. the simple harmonic oscillator.The potential appears in many contexts, e.g. epitaxially-formed quantum dots, secondquantization of the electromagnetic field, vibrations in solids and molecules, MEMS/NEMS, etc,thus it is important to examine this problem in more detail. Let us write out the Hamiltonian fora classical harmonic oscillator and it’s quantum mechanical analogue:

H =p2

2m+

1

2kx2 �! �

~2

2m

d2

dx2+

1

2kx2 = E �!

d2

d⇠2+ (�� ⇠2) = 0

(3)

where ⇠ = ↵x , ↵4 =mk

~2=⇣m!

~

⌘2and ! =

r

k

m, � =

2E

~!



Hermite’s Equation

We can solve equation (3) by substituting solutions of the form: (⇠) = H(⇠)e�⇠2/2, whichyields the following ancillary equation:

d2H

d⇠2� 2⇠

dH

d⇠+ (�� 1)H = 0

this is known as Hermite’s equation, and is solved by assuming a power series solution of theform:

H(⇠) = ⇠s(ao + a1⇠ + a2⇠2 + ...)

putting this into Hermite’s equation and equating powers of ⇠ to zero:

s(s � 1)ao = 0

(s + 1)sa1 = 0

(s + 2)(s + 1)a2 � (2s + 1� �)ao = 0

...

we arrive at a recursion relation:

(s + ⌫ + 2)(s + ⌫ + 1)a⌫+2 = (2s + 2⌫ + 1� �)a⌫

for this series to terminate (and thus be finite), we must have:

� = 2n + 1 where n is an integer, and thus �! En = ~!(n + 1/2)



Eigenfunctions of QHO

The resulting Hn(⇠) are termed Hermite polynomials, and the Eigenfunctions of the quantumoscillator are thus:

n(x) =

✓

↵p⇡n!2n

◆1/2

e�↵2x2/2Hn(↵x) with En = ~!(n + 1/2)

where the coe�cient was worked out by normalization of the wavefunction. At this stage, it isworth noting that the energy Eigenvalues take a distinctly di↵erent form than the previoussolutions, let us look at these graphically:

Figure: Energy Eigenstates of the quantum harmonic oscillator, showing equally-spaced levelssuperimposed on the quadratic potential, including the zero point energy.



Algebraic Solution

As mentioned, the quantum oscillator problem is ubiquitous in nature, and the meaning of theindex n can be assigned to a quantum of energy. For quantized electromagntic fields, we termthese quanta ”photons”, for quantized vibrations in solids or molecules we term them”phonons”. These quantized waves are somewhat unique from other quantum particles, buttheir mathematical treatment is similar. The quantum oscillator problem can be re-cast in anoperator formalism. Deriving this from the above solutions is primarily an algebraic problem,and this is sometimes referred to as the algebraic solution. We begin by calculating theexpectation values of x , x2, and d

dxfor arbitrary n:

Z 1

�1 ⇤n (x)x m(x)dx =

8

>

>

<

>

>

:

1↵

q

n+12 m = n + 1

1↵

q

n2 m = n � 1

0 otherwise

Z 1

�1 ⇤n (x)x

2 m(x)dx =

8

>

<

>

:

2n+12↵2 m = np

(n+1)(n+2)

2↵2 m = n + 20 otherwise

Z 1

�1 ⇤n (x)

d

dx m(x)dx =

8

>

>

<

>

>

:

↵q

n+12 m = n + 1

�↵q

n2 m = n � 1

0 otherwise



Creation and Annihilation Operators

From these results, we can know the expectation values for all the n Eigenstates simply given nand m, so it would be useful to write out the Hamiltonian in a way that would facilitatecalculating these without doing all the integrals. To do this, we define new operators a and a†:

a =↵p2x +

ip2~↵

p =↵p2x +

1p2↵

@

@xand a† =

↵p2x �

ip2~↵

p =↵p2x �

1p2↵

@

@x

with these new operators, we can compute their expecation values in the same way:

Z 1

�1 ⇤n (x) a m(x)dx =

⇢pn + 1 m = n + 10 otherwise

Z 1

�1 ⇤n (x) a

† m(x)dx =

⇢pn m = n � 10 otherwise

Symbollically, we write:

a† n =pn + 1 n+1 and a n =

pn n�1

a and a† are termed the “annihilation” and “creation” operators, respectively, as they lower orraise the energy of the system by moving the particle up or down the “energy ladder”.



The number operator and the QHO Hamiltonian

We can now calculate the commutator of a and a†:

[a, a†] = [↵p2x +

ip2~↵

p,↵p2x �

ip2~↵

p] = �i

2~[x , p] +

i

2~[p, x]

From the above, we can show that [p, x] = �i~ and therefore [a, a†] = 1. Inverting thedefinitions of a and a†:

x =1

p2↵

(a† + a) and p =i~↵p2(a† � a)

we can re-write the Hamiltonian:

H =p2

2m+

1

2kx2 with ↵4 =

mk

~2=⇣m!

~

⌘2and ! =

r

k

mtherefore �! H =

~!2

(aa†+ a†a)

as [a, a†] = 1, aa† = 1 + a†a we can write:

H = ~!(a†a+ 1/2) (4)

using the above definitions of a and a†, we see:

a†a n = a†pn n�1 = n n

for this reason we define: a†a as the number operator n, and re-write the Hamiltonian as:

bH = ~!(n + 1/2)



bra - ket notation and Matrix Formulation of QM

the action of the Hamiltonian on the Eigenfunctions retrieves the Eigenvalues En = ~!(n+1/2):

bH n = En n = ~!(n + 1/2) n

This equation is often written in the the bra-ket notation, where the states are numberedaccording to the energy-ladder shown in figure 2:

bH|ni = ~!(n + 1/2)|ni

here |ni is the “ket” and hn| is the “bra”, the complex conjugate of the “ket”, and theintegrations are implied. Using this notation, we can write expectation values (also called matrixelements) in a compact notation. e.g.:

hn| bH|ni = h bHi = ~!(n + 1/2)

is the expectation value of the energy E . The wavefunction in general is written as asuperposition of these states, as shown in the figure below. This notation leads naturally to therepresentation of the wavefunction as a vector, which in turn leads to Heisenberg’s equivalentmatrix formulation of quantum mechanics, where the Hamiltonian takes a matrix representation :

2

6

6

4

H11 H12 .. H1n

H21 H22 .. H2n

.. .. .. ..Hn1 Hn2 .. Hnn

3

7

7

5

2

6

6

4

c1|1ic2|2ic..|..icn|ni

3

7

7

5

= E

2

6

6

4

c1|1ic2|2ic..|..icn|ni

3

7

7

5



State-Space Representation

This superposition is graphically illustrated as vector in an abstract ”state-space”, sometimescalled Hilbert space.

Figure: Wavefunction represented as a state-vector in a multi-dimensional space using thebra-ket notation.



First Order Perturbation Theory

How does one make a transition from one Eigenstate to another? For this, we need to considerperturbation theory. The assumption is made that a weakly perturbed system can be describedin terms of a linear combination of the unperturbed Eigenstates:

(r, t) =X

n

cn(t) n(r)e�iEnt/~ (5)

We now partition the Hamiltonian into an ”unperturbed” component bHo and the perturbationbHint :

bH = bHo + � bHint (6)

where as before, bHo n = En n is satisfied by the set of Eigenfunctions { n} and their

corresponding Eigenvalues En. bHint refers to the perturbation which disturbs the particle out ofits initial Eigenstate. The parameter � is restricted to � 2 {0, 1}, and represents the strength of

the perturbation, assumed here to be small compared to bHo . We will now insert equations 5 and6 into the time-dependent Schrodinger’s equation bH (r, t) = i~ @

@t (r, t):

X

n

( bHo + � bHint)cn(t) n(r)e�iEnt/~ =X

n

i~cn(t) n(r)e�iEnt/~ +X

n

Encn(t) n(r)e�iEnt/~ (7)



Transition amplitudes

Using the orthogonality of the set of Eigenfunctions { m(r)}, we multiply both sides of equation7 by m(r) and integrate over all space, yielding an equation for each of the cm(t)0s:

i~cm(t)e�iEmt/~ =X

n

cn(t)e�iEnt/~hm|� bHint |ni (8)

Defining !mn = (Em � En)/~, equation 8 can be written:

cm(t) =1

i~X

n

cn(t)hm|� bHint |niei!mn t (9)

Where we have used the bra-ket notation to simplify the resulting expression. Following similarlogic, we can write:

n(r) = (0)n (r) + �

(1)n (r) + �2

(2)n (r) + ...and En = E

(0)n + �E

(1)n + �2E

(2)n + ...

or equivalently,

c(0)n + �c

(1)n + �2c

(2)n + ...

where the superscript represent the ”order”, that is the power in � of the correction being addedto the 0-th order solution, or unperturbed Eigenstate/Eigenfunctions.



First Order Approximation

To first order in �, we would retain only the c(0)n ’s on the righthandside of 9, as � appears as a

common factor in the interaction Hamiltonian for all terms. Equating equal powers of � oneither side of 9 we arrive at the following relations:

c(0)m (t) = 0 (10)

c(s+1)m (t) =

1

i~X

n

c(s)n (t)hm| bHint |niei!mn t (11)

If we assume the perturbation is zero until some time t = 0, we can then make the assumption

that the system was in a pure Eigenstate for t < 0, and thus c(0)i (t < 0) = 1 and all other

c(0)n (t < 0) = 0 for n 6= i . With these assumptions, we can integrate the first order contributionsof equation 11:

c(1)m (t) =

1

i~

Z t

�1hm| bHint |iiei!mi

t (12)



Sinusoidal Perturbation

An bHint of particular relevance is a sinusoidal perturbation, such as an electromagnetic wave:

bHint(t) = bHint(0) sin(!t)

we assume the perturbation is ”on” only for a period of time to , e.g. the lifetime of an excitedstate of a molecule. We could then integrate equation 12:

c(1)m (t > to) = �

hm| bHint(0)|iii~

ei(!mi+!)to � 1

!mi + !�

ei(!mi�!)to � 1

!mi � !

!

(13)

Each of the two terms in equation 13 describe the absorption (!mi = !) or emission (!mi = �!)of a quanta of energy equal to ~!. Let us now assume the former case, in which case we canignore the first term and calculate the probability to arrive in the final state m = f :

|c(1)f (t > to)|2 = 4|hf | bHint(0)|ii|2sin2[ 12 (!fi � !)to ]

~2(!fi � !)2(14)



Transition Rate

To calculate the transition rate, we will need to sum all possible states and divide by the intervalover which the perturbation is non-zero:

Wfi =1

to

X

k

|c(1)k (t > to)|2 �! Wfi =1

to

Z

|c(1)k (t > to)|2⇢(Ek )dEk

where we admit there may be more than one final state, so sum over all final states k, thesecond form has converted the sum into an integral over energy, by introducing the density offinal states ⇢(Ek ) per unit energy, per unit volume. We can assume for many cases, thatto � 1/�!, and as a result this function is very narrow compared to ⇢(Ek ) or the variations in

|hf | bHint(0)|ii|2.

Figure: The function sinc2(1/2(!fi � !)to) compared to the density of states.



Fermi’s Golden Rule

We take ⇢(Ek ) and |hf | bHint(0)|ii|2 out of the integral, and make the change of variablesx = 1

2 (!fi � !)to , the remaining integral over sin2(x)/x2 is tabulated and equal to ⇡, yielding:

Wfi =1

⌧=

2⇡

~|hf | bHint(0)|ii|2⇢(Ef ) �! Wfi =

2⇡

~X

k

|hk| bHint(0)|ii|2�(Eki ± ~!) (15)

The above expressions give the transition rates assuming a sinusoidal perturbation. Therightmost form shows explicitly the sum over all final states and the requirement of energyconservation (the � function), the ±~! includes both the emission and absorption of a photon ofenergy ~!. The above analysis assumes an undetermined amount of energy may be exchangedthrough the perturbation, but in the case of electromagnetic radiation, one must be furtherconcerned with the existence of an electromagnetic mode. In the vicinity of a nanostructure, thenumber of modes at specific frequencies and wavevector may be much larger or smaller thanthat of free space. This is particularly important in the near-field of metallic nano structureswhich may have an optical resonance, e.g. a plasmon resonance or cavity resonance. If weassume bHint = �p · E, then, for a two-level system we may write Fermi’s Golden Rule as:

Wfi = �o |p|2⇢(rm,!) (16)

Where �o is the radiative rate in free space, p is the transition dipole moment, and ⇢(rm,!) isthe local photonic density of states at the location rm at frequency !.



Orthogonality

We wish to show:Z

⇤n0 nd

3r =

⇢

1 n = n0

0 n 6= n0

first multiply above by En:

En

Z

⇤n0 nd

3r =

Z

⇤n0bH nd

3r

and similar for En0 :

En0

Z

⇤n n0d

3r =

Z

⇤nbH n0d

3r

now subtract the two:

Z

n

⇤n0bH n � ⇤

nbH n0

o

d3r = (En�En0 )

Z

⇤n0 nd

3r noting that

Z

⇤n0 nd

3r =

Z

⇤n n0d

3r



Orthogonality ...

Next, insert the form for the Hamiltonian:

Z

⇢

⇤n0

✓

�~2

2mr2 + V

◆

n � ⇤n

✓

�~2

2mr2 + V

◆

n0

�

d3r = (En � En0 )

Z

⇤n0 nd

3r

~2

2m

Z

�

⇤nr2 n0 � ⇤

n0r2 n

d3r = (En�En0 )

Z

⇤n0 nd

3r as:

Z

⇤n0V nd

3r =

Z

⇤nV n0d

3r

Using Green’s identity:R

V (fr2g � gr2f )dv =

R

A(frg � grf ) · nda, we can write this as:

~2

2m

Z

A

�

⇤nr n0 � ⇤

n0r n

· nda = (En � En0 )

Z

⇤n0 nd

3r

The lefthand side goes to zero as A �! 1, as the wavefunction also goes to zero, thus:

0 = (En � En0 )

Z

⇤n0 nd

3r

Evidently, if En 6= En0 , then:

Z

⇤n0 nd

3r =

⇢

1 n = n0

0 n 6= n0⇤



Completeness

We wish to show:X

n

n(r0) n(r) = �(r� r0)

By induction, if:

(r) =X

n

cn n(r), where cn =

Z

(r0) ⇤n (r

0)d3r 0

Then:

(r) =X

n

⇢

Z

(r0) ⇤n (r

0)d3r 0�

n(r)

which we write as:

(r) =Z

d3r 0 (r0)

(

X

n

⇤n (r

0) n(r)

)

=

Z

(r0)�(r� r0)d3r 0

therefore, by inspection:X

n

n(r0) n(r) = �(r� r0) ⇤

The above is also sometimes referred to as the projection operator.


Documents

Theory and Application of Nanomaterialsahrenkiel.sdsmt.edu/courses/Fall2019/NANO702/... · “state-space”, which allow a description of transitions between quantum states. Figure: