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- Solve the TISE for various 1D potentials
- Free particle
- Infinite square well
- Finite square well
- Particle flux
- Potential step
- Transmission and reflection coefficients
- The barrier potential
- Quantum tunnelling
- Examples of tunnelling
- The harmonic oscillator

A Free Particle

Free particle: no forces so potential energy independent of position (take as zero)

Linear ODE with constant coefficients so try

Time-independent Schrödinger equation:

General solution:

Combine with time dependence to get full wave function:

Notes

- Plane wave is a solution (just as well, since our plausibility argument for the Schrödinger equation was based on this assumption).
- Note signs in exponentials:
- Sign of time term (-iωt) is fixed by sign adopted in time-dependent Schrödinger Equation
- Sign of position term (±ikx) depends on propagation direction of wave. +ikx propagates towards +∞ while -ikx propagates towards –∞
- There is no restriction on k and hence on the allowed energies. The states form a continuum.

Particle in a constant potential

General solutions we will use over and over again

Time-independent Schrödinger equation:

Case 1: E > V

(includes free particle with

V = 0 and K = k)

Solution:

Case 2: E < V

(classically particle can not be here)

Solution:

-a

a

Infinite Square WellConsider a particle confined to a finite length –a<x<a by an infinitely high potential barrier

No solution in barrier region (particle would have infinite potential energy).

x

In the well V = 0 so equation is the same as before

General solution:

Boundary conditions:

Continuity of ψ at x = a:

Note discontinuity in dψ/dx allowable, since potential is infinite

Continuity of ψ at x = -a:

Infinite Square Well (2)

Add and subtract these conditions:

Even solution: ψ(x) = ψ(-x)

Odd solution: ψ(x) = -ψ(-x)

Energy

We have discrete states labelled by an integer quantum number

Infinite Square Well (3) Normalization

Normalize the solutions

Calculate the normalization integral

Normalized solutions are

Sketch solutions

Wavefunctions

Probability density

3

1

Note: discontinuity of gradient of ψat edge of well.

OK because potential is infinite there.

Relation to classical probability distribution

Classically particle is equally likely to be anywhere in the box

Quantum probability distribution is

But

so the high energy quantum states are consistent with the classical result when

we can’t resolve the rapid oscillations.

This is an example of the CORRESPONDENCE PRINCIPLE.

Infinite Square Well (5) – notes

- Energy can only have discrete values: there is no continuum of states anymore. The energy is said to be quantized. This is characteristic of bound-state problems in quantum mechanics, where a particle is localized in a finite region of space.
- The discrete energy states are associated with an integer quantum number.
- Energy of the lowest state (ground state) comes close to bounds set by the Uncertainty Principle:
- The stationary state wavefunctions are even or odd under reflection. This is generally true for potentials that are even under reflection. Even solutions are said to have even parity, and odd solutions have odd parity.
- Recover classical probability distribution at high energy by spatial averaging.
- Warning! Different books differ on definition of well. E.g.
- B&M: well extends from x = -a/2 to x = +a/2.

Our results can be adapted to this case easily (replace a with a/2).

- May also have asymmetric well from x = 0 to x = a.

Again can adapt our results here using appropriate transformations.

I

II

III

V0

x

-a

a

i.e. particle is bound

Finite Square WellNow make the potential well more realistic by making the barriers a finite height V0

Region I:

Region II:

Region III:

Finite Square Well (2)

Boundary conditions: match value and derivative of wavefunction at region boundaries:

Match ψ:

Match dψ/dx:

Now have five unknowns (including energy) and five equations

(including normalization condition)

Solve:

Finite Square Well (3)

Even solutions when

Cannot be solved algebraically. Solve graphically or on computer

Odd solutions when

We have changed the notation into q

Finite Square Well (4) Graphical solution

k0 = 4

a = 1

Even solutions at intersections of blue and red curves (always at least one)

Odd solutions at intersections of blue and green curves

Sketch solutions

Wavefunctions

Probability density

Note: exponential decay of solutions outside well

Finite Square Well (6): Notes

- Tunnelling of particle into “forbidden” region where V0 > E
- (particle cannot exist here classically).
- Amount of tunnelling depends exponentially on V0 – E.
- Number of bound states depends on depth of well,
- but there is always at least one (even) state
- Potential is even, so wavefunctions must be even or odd
- Limit as V0→∞:
- We recover the infinite well solutions as we should.

Material B (e.g. GaAs)

Electron energy

Position

Example: the quantum wellQuantum well is a “sandwich” made of two different semiconductors in which the energy of the electrons is different, and whose atomic spacings are so similar that they can be grown together without an appreciable density of defects:

Now used in many electronic devices (some transistors, diodes, solid-state lasers)

Summary of Infinite and Finite Wells

Infinite wellInfinitely many solutions

Even parity

Odd parity

Finite well Finite number of solutions

At least one solution (even parity)

Evanescent wave outside well.

Odd parity solutions

Even parity solutions

Particle Flux

In order to analyse problems involving scattering of free particles, need to understand normalization of free-particle plane-wave solutions.

Conclude that if we try to normalize so that

we get A = 0.

This problem is related to Uncertainty Principle:

Position completely undefined; single particle can be anywhere from -∞ to ∞, so probability of finding it in any finite region is zero

Momentum is completely defined

Solutions: Normalize in a finite box

Use wavepackets (later)

Use a flux interpretation

a

b

Particle Flux (2)More generally: what is the rate of change of probability that a particle is in some region (say, between x=a and x=b)?

Use time-dependent Schrödinger equation:

a

b

Particle Flux (3)Flux entering at x=a

minus

Flux leaving at x=b

Interpretation:

Note: a wavefunction that is real carries no current

Particle Flux (4)

Check: apply to free-particle plane wave.

Makes sense:

# particles passing x per unit time = # particles per unit length × velocity

So plane wave wavefunction describes a “beam” of particles.

- Time-independent case: replace
- 3D case,
- Can use this argument to prove CONSERVATION OF PROBABILITY.
- Put a = -∞, b = ∞, then

and

Particle Flux (5): NotesV0

Case 1

0

x > 0, V =

V0

Potential StepConsider a potential which rises suddenly at x = 0:

x

Boundary condition: particles only incident from left

Case 1: E > V0 (above step)

x < 0, V = 0

Potential Step (3) Transmission and Reflection Fluxes

Calculate transmitted and reflected fluxes

x < 0

x > 0

(cf classical case: no reflected flux)

Check: conservation of particles

V0

0

Potential Step (4)Case 2: E < V0(below step)

Solution for x < 0same as before

Solution for x > 0is now evanescent wave

Matching boundary conditions:

Transmission and reflection amplitudes:

Transmission and reflection fluxes:

This time we have total reflected flux.

Some tunnelling of particles into classically forbidden region even for energies below step height (case 2, E < V0).

- Tunnelling depth depends on energy difference
- But no transmitted particle flux, 100% reflection, like classical case.
- Relection probability is not zero for E > V0 (case 1). Only tends to zero in high energy limit, E >> V (correspondence principle again).

I

II

III

V0

0

b

Rectangular Potential BarrierNow consider a potential barrier of finite thickness:

x

Boundary condition: particles only incident from left

Region I:

Region II:

Region III:

u = exp(ikx) + B exp(−ikx)

u = C exp(Kx) + D exp(−Kx)

u = F exp(ikx)

Rectangular Barrier (2)

Match value and derivative of wavefunction at boundaries:

Match ψ:

1 + B = C + D

1 − B = K/(ik)(C − D)

C exp(Kb) + D exp(−Kb) = F exp(ikb)

C exp(Kb) − D exp(−Kb) = ik/K F exp(ikb)

Match dψ/dx:

Eliminate wavefunction in central region:

Rectangular Barrier (3)

Transmission and reflection amplitudes:

F

For very thick or high barrier:

|F|2 =

Non-zero transmission (“tunnelling”) through classically forbidden barrier region. Exponentially sensitive to height and width of barrier.

Repulsive Coulomb interaction

Incident particles

Assume a Boltzmann distribution for the KE,

Probability of nuclei having MeV energy is

Nuclear separation x

Strong nuclear force (attractive)

Examples of TunnellingTunnelling occurs in many situations in physics and astronomy:

1. Nuclear fusion (in stars and fusion reactors)

Fusion (and life) occurs because nuclei tunnel through the barrier

Distance of α-particle from nucleus

Initial α-particle energy

Examples of Tunnelling2. Alpha-decay

α-particle must overcome Coulomb repulsion barrier.

Tunnelling rate depends sensitively on barrier width and height.

Explains enormous range of α-decay rates, e.g. 232Th, t1/2 = 1010 yrs, 218Th, t1/2 = 10-7s.

Difference of 24 orders of magnitude comes from factor of 2 change in α-particle energy!

x

Probe

Material

a

Vacuum

Examples of Tunnelling3. Scanning tunnelling microscope

A conducting probe with a very sharp tip is brought close to a metal. Electrons tunnel through the empty space to the tip. Tunnelling current is so sensitive to the metal/probe distance (barrier width) that even individual atoms can be mapped.

Tunnelling current proportional to

and

so

If a changes by 0.01A (~1/100th of the atomic size) then current changes by a factor of 0.98,

i.e. a 2% change, which is detectable

STM image of Iodine atoms on platinum.

The yellow pocket is a missing Iodine atom

Summary of Flux and Tunnelling

The particle flux density is

Particles can tunnel through classically forbidden regions.

Transmitted flux decreases exponentially with barrier height and width

We get transmission and reflection at potential steps.

There is reflection even when E > V0.

Only recover classical limit for E >> V0 (correspondence principle)

x

V(x)

x

Simple Harmonic OscillatorExample: particle on a spring, Hooke’s law restoring force with spring constant k:

Time-independent Schrödinger equation:

Problem: still a linear differential equation but coefficients are not constant.

Simplify: change to dimensionless variable

Simple Harmonic Oscillator (2)

Asymptotic solution in the limit of very large y:

Try it:

Equation for H(y):

Simple Harmonic Oscillator (3)

Solve this ODE by the power-series method (Frobenius method):

Find that series for H(y)must terminate for a normalizable solution

Can make this happen after n terms for either even or odd terms in series (but not both) by choosing

Hence solutions are either even or odd functions (expected on parity considerations)

0

Label normalizable functions H by the values of n (the quantum number)

Hnis known as the nth Hermite polynomial.

Simple Harmonic Oscillator (4)

EXAMPLES OF HERMITE POLYNOMIALS

AND SHO WAVEFUNCTIONS

are normalization constants

Simple Harmonic Oscillator (5) Wavefunctions

High n state (n=30)

wavefunction

Probability density

- Decaying wavefunction tunnels into classically forbidden region
- Spatial average for high energy wavefunction gives classical result:
- another example of the CORRESPONDENCE PRINCIPLE

1) The quantum SHO has discrete energy levels because of the normalization requirement

2) There is ‘zero-point’ energy because of the uncertainty principle.

3) Eigenstates are Hermite polynomials times a Gaussian

4) Eigenstates have definite parity because V(x) = V(-x). They can tunnel into the classically forbidden region.

5) For large n (high energy) the quantum probability distribution tends to the classical result. Example of the correspondence principle.

6)Applies to any SHO, eg: molecular vibrations, vibrations in a solid (phonons), electromagnetic field modes (photons), etc

Summary of Harmonic Oscillator

Example of SHOs in Atomic Physics:

Bose-Einstein Condensation

87Rb atoms are cooled to nanokelvin temperatures in a harmonic trap. de Broglie

waves of atoms overlap and form a giant matter wave known as a BEC. All the atoms go into the ground state of the trap and there is only zero point energy (at T=0). This is a superfluid gas with macroscopic coherence and interference properties.

Signature of BEC phase transition:

The velocity distribution goes from classical Maxwell-Boltzmann form to the distribution of the quantum mechanical SHO ground state.

V(x)

x

H

H

Nuclear separation x

SHO levels

Example of SHOs: Molecular vibrations

VIBRATIONAL SPECTRA OF MOLECULES

Useful in chemical analysis and in astronomy

(studies of atmospheres of cool stars and interstellar clouds).

SHO very useful because any potential is approximately parabolic near a minimum

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