Quantum Mechanics IB

7/22/2019 Quantum Mechanics IB

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Class 2 Quantum

mechanics-IBDr. Marc Madou Chancellor’s Professor

UC Irvine, 2012

2(x,t)

x2

8 2m

h2

E - V(x,t) (x,t) 0



Contents Time Independent SE or TISE

Applying Schrödinger’s Equation:

Particles in a Large (Infinite) Solid Particles in a Finite Solid

Quantum Wells (1D confinement)-DOS

Quantum Wires (2D confinement)-DOS

Quantum Dots (3D confinement)-DOS

Tunneling Harmonic Well

Central Force



Time Independent SE or TISE

In the case V(x, t) is independent of time, the SE can beconverted into a time independent SE (TISE). Hence we obtainthe time independent form of the Schrödinger’s equation as:

Solving this equation, say for an electron acted upon by a fixednucleus, we will see that this results in standing waves.

The more general Schrödinger equation does feature a time

dependent potential V=V(x,t) and must be used for examplewhen trying to find the wave function of say an atom in aoscillating magnetic field or other time-dependent phenomenasuch as photon emission and absorption.

2(x,t)

x2

8 2m

h2

E - V(x,t) (x,t) 02 2

2V(x) (x)=E (x)

2m x




Particles in a Large (Infinite) Solid For free electrons in an infinitely large 3D piece of

metal the allowed electron states are solutions of an expanded version of the Schrödinger equation :

For electrons swarming around freely in thisinfinite metal, the potential energy V(r) is zeroinside the conductor and the solutions inside themetal are plane waves moving in the direction of r:

where r is any vector in real space and k is any wavevector. As with a freely moving particle,normalization is impossible as the wave extends toinfinity.

2

(r)

8 2m

h2 E -V(r) (r) 0

k (r) Aexpik r




Particles in a Large (Infinite) Solid

Plotting the energy E versus the wavenumber k x for a free electron gas leads to a

parabolic dispersion relation.

From classical theory we could notappreciate the occurrence of long electronicmean free paths, indeed Drude used the

interatomic distance “a” for the mean free path . But from experiments with very purematerials and at low temperatures it is clear the mean free path may be much longer,actually it may be as long as 108 or 109 interatomic spacings or more than 1 cm.

The quantum physics answer is that theconduction electrons are not deflected by ioncores arranged in a periodic lattice becausematter waves propagate freely through a

periodic structure just as predicted by:

k (r) Aexpik r

Confining electrons by

limiting their propagation

in certain directions in acrystal introduces a

varying V(r) in the

Schrödinger’s equation

and this may lead to an

electronic band gap.

2 (r)8 2m

h2E - V(r) (r) 0




Particles in a Finite Solid

Discrete energy levels inevitably arisewhenever a small particle such as a photon or electron is confined to a region in space.

Sommerfeld, in 1928, was the first to show this.He adopted Drude’s free electron gas (FEG)and added the restriction that the electrons must

behave in accordance with the rules of quantummechanics (e.g., only 2 electrons per energylevel) or he defined a Fermi gas.

In his Fermi gas, electrons are free, except for their confinement within a cubic piece of crystalline conductor with a finite volume of V=L3 and they follow Fermi-Dirac statisticsinstead of Maxwell-Boltzmann rules.




Particles in a Finite Periodic Solid Outside the 3D cube of solid the potential V(x)= and

the wave function is zero anywhere outside the solid.This situation applies, for example, to totally freeelectrons in a metal where the ion cores do not influencetheir movement. Sommerfeld actually assumed that V(x)outside the conductor equaled the work function .

Now let’s introduce periodicity e.g. a repeating cube withside L. The choice of a cube shape with side L is amatter of mathematical convenience. The value L is set

by the Born and von Karman’s periodic boundarycondition, i.e., that the wave functions must obey thefollowing rule: (x+L, y+L, z+L)=(x,y,z)

Setting such a periodic boundary condition ensures thatthe free-electron form of the wave function is NOTmodified by the shape of the conductor or its boundary.This can be interpreted as follows: an electron coming tothe surface is not reflected back in, but reenters the metal

piece from the opposite surface. This excludes thesurfaces from playing any role in transport phenomena.




Particles in a Finite Periodic Solid with

Varying Vr Besides periodicity we also introduceV( r ) ≠ 0. The electrons are not totally

free anymore ! They feel the ion cores.

Energy versus wave number for motion of

an electron in a one-dimensional periodic

potential. The range of allowed k valuesgoes from –π/a to + π/a corresponding to

the first Brillouin zone for this system.

Similarly, the second Brillouin zone

consists of two parts; on extending from

π/a to 2π/a, and another part extending between -π/a and -2π/a. This

representation is called the extended zone

scheme. Deviations from free electrons

parabola are easily identified. Where a is

the lattice constant.




Quantum Wells (1D confinement)

Outside the well and = 0 for

For the Schrödinger equation inside the

material (0 <x < L) we write:

V

x 0 and x L

2

2me

d2 x dx2

E x

n (x, t) =

2

Lsin

n x

L

En (k ) =2

2me

(nL

)2





At the lowest energy (n=1),the ground state, the energyremains finite despite the factthat V=0 inside the region.According to quantummechanics an electron cannot

be inside the box and havezero energy. This is called thezero-point energy animportant consequence of theHeisenberg principle.





For the same value of quantum number n,the energy is inversely proportional to themass of the particle and to the square of the length of the box. For a heavier

particle and a larger box, the energy levels

become more closely spaced. Only whenmL2 is of the same order as , doquantized energy levels become importantin experimental measurements (with L =1nm, ). With a 1 cm3 pieceof metal (instead of 1nm3), the energy

levels become so closely spaced that theyseem to be continuous ,in other words the quantum mechanicsformula gives the classical result for dimension such that meL

2 >>2

2

E1 h

2

8meL2 0.36eV

Zhores Alferov (Left) and

Herbert Kroemer (Right).Nobel Prize in Physics 2000.

E1

h2

8meL2

3.6 1015eV





QWs developed in the early 1970s and constituted thefirst lower dimensional hetero-structures. Theforemost advantage of such a design involves their improved optical properties.

In a quantum well there are no allowed electron statesat the very lowest energies (an electron in a box withenergy = 0 does not exist) but there are many moreavailable states (higher DOS) in the lowest conductionstate so that many more electrons can beaccommodated. Similarly, the top of the valence band

has plenty more states available for holes. This meansthat it is possible for many more holes and electrons tocombine and produce photons with identical energyfor enhanced probability of stimulated emission(lasing)



DOS-Bulk Materials

We calculate for the density of states for a parabolic

band in a bulk material (3 degrees of freedom) as:

G(E)3DdE 1

2 22me

*

2

3/2

E1

2dE



DOS-Quantum Well The density of states function (DOS) for a quantum well is different from that of a 3D

solid. The solid black curve is that for free electrons in all 3 dimensions. The bottom of thequantum well is at energy Eg but the first level is at E0. This causes a blue shift.

There are many more states at E0 than at Eg.This makes, for example, for a better laser.

To calculate the current density for a 2D electron gas at a particular temperature we firstcalculate the value for n(E)2D at T > 0.The function n(E)2D for a given temperature T (>0)is shown as a red line. In the same graph we also show the Fermi-Dirac function and the

DOS function (blue).

J = -n(E)2Devavg

n(E)2DdE =G(E)2Df(E)dE




Quantum Wires (2D confinement)

For a particle in a finite sized, 2-D

infinitely deep potential well, we define

a wave function similar to the 1-D potential well, but now we obtain x,y)

solutions that are defined by 2 quantum

numbers one associated with each

confined dimension.




Quantum Wires (2D confinement) Compared to the fabrication of

quantum wells, the realization of

nanoscale quantum wires requires

more difficult and precise growth

control in the lateral dimension, and,

as a result, quantum wire

applications are in the development

stage only.

NEC’s Sumio Iijima

1 2

1

21 2

n n

1 2 1 2

n x n y4(x,y)= sin sin

L L L L

1 2 1 2

2 22x y 1 2

n n n n 2 2

1 2

n nhE ( )=E E

8m L L

k



DOS-Quantum Wire Quantum wires again feature a blue

shift.

Also the density of states (DOS) (blue)and occupied states (red) for a

quantum wire are different than those

of a 3D electron gas. The Fermi-Dirac

function is f(E) and n(E)1D is the

product of f(E) and G(E)1D at T>0.

n(E)1DdE =G(E)1Df(E)dE

J = -n(E)2Devavg



DOS-Quantum Wire For a quantum wire the resistance or current is

found to further simplify to a very simple

expression that does not depend on voltage but

only on the number of available levels. The amount of current is dictated only by the

number of modes (also called sub-bands or

channels) M (EF), that are filled

betweenEF(1)andF with each mode

contributing 2e2

/h or:

and :

This quantized resistance R has a value of 12.906

2

F2e M(E )h

2

F

h 1R

2e M(E )



DOS-Quantum Wires Each of the discrete peaks in the density of

states (DOS) is due to the filling of a newlateral sub-band. The peaks in the density of states functions at those energies where the

different sub-bands begin to fill are calledcriticalities or Van Hove singularities. Thesesingularities (sharp peaks) in the density of states function leads to sharp peaks inoptical spectra and can also be observeddirectly with scanning tunneling microscopy

(STM). Singularities again emerge when dk /dE = 0




Quantum Dots (3D confinement)

In the early 1980s, Dr. Ekimov

discovered quantum dots with his

colleague, Dr. Efros, while working

at the Ioffe Institute in St.

Petersburg (then Leningrad), Russia.This team’s discovery of quantum

dots occurred at nearly the same

time as Dr. Louis E. Brus a

physical chemist then working at

AT&T Bell Labs found out how to

grow CdSe nano crystals in a

controlled manner.

Columbia’s Louis Brus




Quantum Dots (3D confinement)

The solution of the SE for a

square semiconductor

quantum dot (3D

confinement) with side =Land volume V =L3 is given as:

k (r) V

1

2exp(ik r)

k x 2 nx

L, k y

2 ny

L, k z

2 nz

L, nx, ny, nz 1,2,3,...

1 2 3

22 22 22 2 231 2

n ,n ,n 1 2 32 2 2 2

1 2 3

nn nh hE = = n n n

8m L L L 8mL



DOS Quantumdot

A quantum dot (QD) is an atom-likestate of matter often referred to as an“artificial atom.’

What is so interesting about a QD is thatelectrons trapped in them arrangethemselves as if they were part of anatom although there is no nucleus for the electrons to surround here.

The type of atom the dot emulates

depends on the number of atoms in thewell and the geometry of the potentialwell V(r) that surrounds them.



Conclusion Reduced DOS as a

Function of Dimensionality An important consequence of decreasing the dimensionality beyond that

of quantum wells and quantum wires is that the density of states (DOS)for quantum dots features an even sharper and yet more discrete DOS.

As a consequence, quantum dot lasers exhibit a yet lower thresholdcurrent than lasers based on quantum wire and wells, and because of themore widely separated discrete quantum states they are also lesstemperature sensitive.

However since the active lasing material volume is very small inquantum wires and dots, a large array of them has to be made to reach a

high enough overall intensity. Making quantum wires and dot arrays witha very narrow size distribution to reduce inhomogeneous broadeningremains a real manufacturing challenge and as a result only quantum welllasers are commercially mature.




Tunneling Outside the box, in regions I and III, the

boundary condition is that V(x) = Vo.These are regions that are “forbidden” to

classical particles with E < Vo. With E< V0 a classical particle cannot penetratea barrier region: think about a particlehitting a metal foil and only penetratingthe foil if its initial energy is greater thanthe potential energy it would possess

when embedded in the foil and whereotherwise it will be reflected.

The solution inside the well is anoscillating wave just as in the case of thewell with infinite walls.

2

2m

d2 x dx2

E V0 x




Tunneling Defining as :

Yields:

In a region with E < V0 there isan immediate effect on thewaveform for the particle

because, k x is real under these

conditions and we can write:

2 2m(V0 E)2

d2

dx2 2

k x = =2m(V0 E)




Tunneling We find for the general solution in RegionI and III, a wave-function of the form ,

i.e., a mixture of an increasing and a

decreasing exponential function.

With a barrier that is infinitely thick wecan see that the increasing exponentialmust be ruled out as it conflicts with theBorn interpretation because it would

imply an infinite amplitude. Therefore in a barrier region the wave-function mustsimply be the decaying exponential . Theimportant point being that a particle may

be found inside a classically forbiddenregion (Region I and III).

x Ae x

+Be x




Tunneling The exponential decay of the wave

function inside the barrier is given as:

If the barrier is narrow enough (L)there will be a finite probability P of

finding the particle on the other side of the barrier. The probability of anelectron reaching across barrier L is:

where A is a function of energy E and barrier height V0.

(x) = Ae - x

P | (x) |2 = A2e-2 L




Tunneling The tunneling current, picked up

by the sharp needle point of an

STMis given by:

Where f w(E) is the Fermi-Diracfunction, which contains a

weighted joint local density of

electronic states in the solid

surface that is being probed andthose states in the needle point.

I = f w(E)A2e-2 L




Harmonic Well The time independent

Schrödinger equation

(TISE) for a harmonicoscillator is given as:

2

2m

d2 x dx2

E 12k x2 x

n(x) N

nH

n(

12x)e

- x

2

2

with n 0,1,2,3...and (K m

2)

1

2

Nn (a normalization constant) =1

(2n n!)1

2

(

)

1

4

En n 12 , n 0,1,2,3...




Harmonic Well As expected from Bohr’s

Correspondence principle the

higher the quantum numbers

the better the quantizedoscillator resembles the

classical non-quantized

oscillator.




Central Force An electron bound to the

hydrogen nucleus is an example

of a central force system: the

force depends on the radialdistance between the electron

and the nucleus only.

The solutions of the

Schrödinger equation with this potential are spherical Bessel

functions.



Summary SE

Applications Summarizing, the quantization

for three of the most important

potential profiles leads to the

following mathematicalsolutions of the Schrödinger

equation: for the central force

we obtain spherical Bessel

functions, for an infinite

square well potential sines,

cosines and exponentials and

for an oscillator Hermite

polynomials.



Quantum Jokes

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