Lecture 14 special This lecture will NOT be examined as part of PHY226 although the solutions of the IPW and FPW are excellent examples of the solving

Lecture 14 specialLecture 14 special

This lecture will This lecture will NOTNOT be examined as part of be examined as part of PHY226 although the solutions of the IPW and PHY226 although the solutions of the IPW and FPW are excellent examples of the solving of FPW are excellent examples of the solving of

ODEs covered in lecture 3ODEs covered in lecture 3

Lecture 14 specialLecture 14 special

An introduction to SchrAn introduction to Schrödingerdinger

http://www.hep.shef.ac.uk/Phil/PHY226.htmRemember Phils Problems and your notes = everything

t

uiVuu

m

22

2

Introduction to Quantum MechanicsIntroduction to Quantum MechanicsSomething weird is going on……justification of Quantum Mechanics

If you were to shine white light through sodium gas you would see an absorption line in the total spectrum.

Spectral lines come from the fact that atoms absorb or emit particular energies of photons.

Passing a current through sodium gas causes the emission of monochromatic light

Sodium street light


Hydrogen gas for example can therefore only absorb or emit specific wavelengths of light. When a photon is absorbed an electron is promoted to a higher energy level, but only specific energy photons can do this, it follows that discrete energy levels exist in the atom.

This provides us with a spectral fingerprint that tells us about the inner structure of the atom.

Looking at the continuous spectrum and comparing it with the Hydrogen emission spectrum we see that the five emission lines are at 1.9, 2.5, 2.8, 3.0, 3.1eV.

E=hf


The energy levels are said to be quantised and when the electron moves from one energy level to another they are said to make quantum jumps.

For example when a current is passed through sodium gas, electrons are promoted to higher energy levels, they then drop to lower energy levels and emit discrete energies or wavelengths of light.

The energy of the photons emitted are equal to the differences between the energy levels so the 2.1eV photons emitted by street lights tell us that there is an energy gap between levels of 2.1eV.


The usual position of the electron in the hydrogen atom is in the ground state E1 whereas E∞ corresponds to the situation where the electron is separated from the proton by an infinite amount.

E∞ - E1 corresponds to the ionisation energy which is 13.6eV for Hydrogen.

Once the electron has been ionised, any remaining energy increases the kinetic energy of the proton and electron which, existing in the continuum region can have any value of energy.

five emission lines are at 1.9, 2.5, 2.8, 3.0, 3.1eV

NB. The energy of the emission line is determined by ΔE (the difference in energy states)

Although there are 5 visible lines there are many more in the EM spectrum

Introduction to Quantum MechanicsIntroduction to Quantum Mechanics

Lots of people tried to explain atomic structure before quantum mechanics came along.

Prediction of atomic structure

The electrical attraction held them together and the rotational motion kept them apart. In the model the electron can exist at any radius so long as the rotational velocity increases to compensate.

Rutherford’s atom had a central positive nucleus with a negative electron orbiting it.

This model therefore does not predict discrete energy levels or explain line spectra.


Prediction of atomic structure

Bohr then developed the Newtonian and electromagnetic ideas of electron motion and managed to construct a Hydrogen atom with the discrete energy levels required for line emission.

However when an electron rotates it constantly undergoes acceleration and emits electromagnetic radiation. Therefore we would expect the electron to constantly emit photons, losing energy constantly and eventually spiralling into the nucleus. This doesn’t happen!!

The model combines classical orbiting electron with quantized electron momenta. The electrons therefore still orbit the nucleus but in this model they are restricted to specific radii.

Introduction to Quantum MechanicsIntroduction to Quantum MechanicsQuantum indeterminacy

If you bombard a hydrogen atom with photons of high enough energy to promote the electron from E1 to E3 then sometimes it will do this and other times it wont !!!

The same occurs for an electron in an excited state that can either drop down one or more energy levels.

We can never know if an individual atom has absorbed a photon or not and the best we can do based on statistics is to assign a probability to whether or not the process will occur.

It predicted the energy levels of the H atom.

Introduction to Quantum MechanicsIntroduction to Quantum MechanicsSchrödinger equation

Both the Rutherford and Bohr models of the atom are therefore flawed.

In the 1920s a group of Physicists headed by Schrodinger developed what we now know as the Schrodinger equation. The equation did two main things.

But it also introduced the concept that the behaviour of the electron is intrinsically indeterminate.

According to Quantum mechanics, if a H atom has a certain amount of energy it is impossible to say in advance of the measurement what value will be obtained for the electrons position and momentum.

This means that if we perform identical measurements on an atom with the same energy, we will always have different outcomes. What can be predicted are the range of possible outcomes of the measurement and the probability of each of these outcomes.


A good way of illustrating the uncertainty of the position of the electron is to show it in 2D as an electron cloud.

The greater the concentration of dots, the more likely the electron would be found at this location. The maximum concentration of dots corresponds to a radius of 5.29×10-11m which is exactly the radius predicted by the Bohr model when the electron is in its ground state.

If we imagine that for a single atom we measure the position of the electron 1000 times and place a single dot on the paper at each location we found the electron to be then we would end up with an electron probability cloud.


So far we’ve only talked about the ground state but the Schrodinger equation can be used to predict the behaviour of the electron in any of its energy states (eigenvalues).

The figure right shows the probability density clouds of the ground state of the H atom and also other higher energy states.

The use of electron probability clouds to predict the probability associated with measuring in the quantum world is visually very clear but nowhere near as useful as the eigenfunctions we are now so familiar with.

But is this really the best way we can represent the likelihood that an electron will be in a certain position at a certain time?

Introduction to Quantum MechanicsIntroduction to Quantum MechanicsThe electron probability cloud is analogous to the probability density function given below, expressing the probability P of finding the particle at some particular location between b and a.

The position of the particle is, of course, one quantity we might imagine measuring experimentally. It is an observable quantity.

bx

ax

Pdxtxtx ),(),(*

1),(),(* dxtxtxRemembering that

But there are many physical observables. One is the energy, which we determine from the solutions of the TISE.

)()()()(2 2

22

xExxVxdx

d

m

We usually see it written like this:

OK, so I know that I have to start thinking about operators and eigenfunctions and eigenvalues but

where do we start ????!!

What you have to realise is that because there is a probability associated with pinning the particle down to a specific energy or position or momentum, with every calculation we need to incorporate the probability associated with the measurement.

You are most familiar with using the TISE to find the specific energy levels associated within a zero potential well where:

LxforxEdx

xd

m

0)(

)(

2 2

22

The whole point of Quantum mechanics !!!!!!!!!

Quantum mechanics does not explain how a quantum particle behaves. Instead, it gives a recipe for determining the probability of the measurement of the value of a physical variable (e.g. energy, position or momentum). This information enables us to calculate the average value of the measurement of the physical variable.

We do this using OPERATORS such as the one below…..

)()( xExH where H denotes the Hamiltonian operator )(2 2

22

xVdx

d

mH

An equation of the form is called an eigenvalue equation.

The values of are called the eigenvalues, and the corresponding functions

are called the eigenfunctions.

Introduction to Quantum MechanicsIntroduction to Quantum MechanicsOperators play a crucial role in the theory of quantum mechanics, as each experimental observable is associated with an operator. A “hat” usually denotes operators.

)()( xx nnnO

n )(xn

The allowed values of the observable are the eigenvalues of the operator, each corresponding to a function (the eigenfunction) which represents the state of the system when the observable has that value.

The TISE is such an equation. The allowed values of energy are the eigenvalues of the Hamiltonian operator, and the corresponding wavefunctions are its eigenfunctions.

)()( xExH

L

xn

Lxn

sin

2)(

Eigenvalue equation Eigenfunction Eigenvalue

2

222

2 L

n

mEn

Introduction to Quantum MechanicsIntroduction to Quantum MechanicsLet’s show how we can find the eigenvalues of energy in zero V using an operator ….

)()( xExH

L

xn

Lxn

sin

2)(

Eigenvalue equation Eigenfunction

Eigenvalue

2

222

2 L

n

mEn

Operator

)(2 2

22

xVdx

d

mH

L

xn

LE

L

xn

Ldx

d

L

n

mH

sin

2cos

2

2

2

L

xn

LE

L

xn

Ldx

d

mH

sin

2sin

2

2 2

22

L

xn

LE

L

xn

LL

n

m

sin

2sin

2

2 2

222

So as expected nEL

n

m

2

222

2

Introduction to Quantum MechanicsIntroduction to Quantum MechanicsThe list of operators is given below:

If we measure the momentum of the momentum eigenfunction, L

ikxx

)exp()(

The operator is, and so dx

d

ip

)()( xpxdx

d

i

)()exp()exp()exp( xpikxL

kikxik

Liikx

dx

d

Li

)exp()exp( ikxL

pikx

L

k

kp so Here p is the eigenvalue, and ψ(x) is the eigenfunction


Infinite potential well

LxforxV 00)(

otherwisexV )(

00)( xforx

Lxforx 0)(

LxforxEdx

xd

m 0)(

)(

2 2

22

The particle cannot exist where the potential is infinite, so the boundary conditions are:

Re-arrange as )(2)(

22

2

xmE

dx

xd

)()( 2

2

2

xkdx

xd 2

2 2

mE

k where and write

As always set so and so mxe )()( 2

2

2

xmdx

xd

22 km

General solution is

ikm

)cos()sin()( kxBkxAx Boundary conditions are

000)( Bsoxatx nkLsokLAsoLxatx 0)sin(0)(


Infinite potential well Thus the solutions are

L

xnAxn

sin)(

The energies are given by 2

2 2

mE

k where 2

22222

22 L

n

mm

kE nn

as

L

nkn

eigenfunctions eigenvalues

Notice how as a consequence of the boundary conditions on ψ(x) at x = 0 and L we must fit an integral number of half-wavelengths into the potential well of width L


Infinite potential well

For each eigenfunction the probability of finding the particle in

the well is unity. Thus,

L

xnAxn

sin)(

1),(),(* dxtxtx

This determines A: so

L

dxL

xnA

0

22 1sin

L

dxL

xnA

0

2 12

cos12

1

12

sin22

1

0

2

L

L

xn

n

LxA

1

2

2

LA

LA

2

so so

and therefore

L

xn

Lxn

sin

2)(So the eigenfunctions and their corresponding eigenvalues

2

22222

22 L

n

mm

kE nn

have been found using the kinetic energy operator


Finite potential well

220)(

Lx

LforxV

otherwiseVxV 0)(

Eigenfunctions with energy eigenvalues E > V0 are unbound Eigenfunctions with energy eigenvalues E < V0 are bound.

For bound states the wavefunction penetrates the classically forbidden region. Thus, the particle exists in a region where its kinetic energy is negative.

To find energies of these states we’ll solve the time independent Schrödinger equation:

Notice that E = KE + PE, thus KE(x) = E - PE = E -V(x)



220)(

Lx

LforxV

otherwiseVxV 0)(

We need to solve:

22)(

)(

2 2

22 Lx

LforIregioninxE

dx

xd

m

REGION I

In region I solutions are )cos()( kxAxI

Using same technique as for infinite well but for different boundary conditions

NB. The value k above is actually smaller than the corresponding value k for the inifinite potential well.

This means that is not 0 at the potential well boundaries.

Also since the energy levels are lower compared to IPW

)(xI

m

kE nn 2

22

IPW

FPW



220)(

Lx

LforxV

otherwiseVxV 0)(

We need to solve:

REGION II

Re-arrange to

2)()(

)(

2 02

22 LxforIIregioninxExV

dx

xd

m

)()(2)(

20

2

2

xEVm

dx

xd

)()( 2

2

2

xdx

xd

0)(2

202

EVm

As always set so and so mxe )()( 2

2

2

xmdx

xd

22 m m

where

The general solution is then xx DeCex )(



REGION II

Boundary conditions xx DeCex )(

to satisfy max probability of 1, thus C = 0. xasx 0)(

must be continuous at boundary between regions I and II at x = L/2. dx

xdandx

)()(

)2()2( LL III

2exp

2cos

LD

kLA

so

dx

L

dx

L III )2()2(

2exp

22sin2

LD

kLkA

so

Dividing eqn (ii) by eqn (i) we eliminate A and D to obtain the condition

(i)

(ii)

k

kL

2

tan


Finite potential well summary

xII Dex )(

k

kL

2

tan

REGION II

REGION I )cos()( kxAxI

If we wanted to we could then perform the same procedure for the region to the left of the potential well.

We would then be able to normalise the function between ± ∞ to unity and find another expression linking the coefficients A and D

1),(),(* dxtxtx

1/α is defined as the penetration depth


Potential wells


Quantum Tunnelling

The same system in quantum mechanics gives a non-zero probability that the particle will be transmitted through the barrier. This is a wave phenomenon, but in quantum mechanics particles exhibit wave-like properties.

The wavefunction of the tunneling particle decreases exponentially in the barrier. The tunneling probability is strongly dependent on the width of the barrier, the mass of the particle, and the quantity (V-E). For instance, the ratio of tunneling probability for protons to electrons is around a factor of 10-91.

Classically, if you have a potential barrier of height V and a particle incident on that barrier with E < V, the particle would reflect off the barrier completely.


Quantum Tunnelling: uses

The most important applications of quantum tunnelling are in semiconductor and superconductor physics.

Phenomena such as field emission, important to flash memory, are explained by quantum tunnelling.

Another major application is in scanning tunnelling microscopes which can resolve objects that are too small to see using conventional microscopes, overcoming the limiting effects of conventional microscopes (optical aberrations, wavelength limitations) by scanning the surface of an object with tunnelling electrons.

Documents

Lecture 14 special This lecture will NOT be examined as part of PHY226 although the solutions of the IPW and FPW are excellent examples of the solving