1D systems. 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.
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Free particle: no forces so potential energy independent of position (take as zero)
Linear ODE with constant coefficients so try
Time-independent Schrödinger equation:
Combine with time dependence to get full wave function:
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)
Case 2: E < V
(classically particle can not be here)
aInfinite Square Well
Consider 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).
In the well V = 0 so equation is the same as before
Continuity of ψ at x = a:
Note discontinuity in dψ/dx allowable, since potential is infinite
Continuity of ψ at x = -a:
Add and subtract these conditions:
Even solution: ψ(x) = ψ(-x)
Odd solution: ψ(x) = -ψ(-x)
We have discrete states labelled by an integer quantum number
Normalize the solutions
Calculate the normalization integral
Normalized solutions are
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
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.
Our results can be adapted to this case easily (replace a with a/2).
Again can adapt our results here using appropriate transformations.
Boundary conditions: match value and derivative of wavefunction at region boundaries:
Now have five unknowns (including energy) and five equations
(including normalization condition)
Even solutions when
Cannot be solved algebraically. Solve graphically or on computer
Odd solutions when
We have changed the notation into q
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
Note: exponential decay of solutions outside well
Material B (e.g. GaAs)
PositionExample: the quantum well
Quantum 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)
Infinite wellInfinitely many solutions
Finite well Finite number of solutions
At least one solution (even parity)
Evanescent wave outside well.
Odd parity solutions
Even parity solutions
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
Check: apply to free-particle plane wave.
# particles passing x per unit time = # particles per unit length × velocity
So plane wave wavefunction describes a “beam” of particles.
Continuity of ψ at x = 0:
Solve for reflection and transmission amplitudes:
Calculate transmitted and reflected fluxes
x < 0
x > 0
(cf classical case: no reflected flux)
Check: conservation of particles
Some tunnelling of particles into classically forbidden region even for energies below step height (case 2, E < V0).
Match value and derivative of wavefunction at boundaries:
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)
Eliminate wavefunction in central region:
Transmission and reflection amplitudes:
For very thick or high barrier:
Non-zero transmission (“tunnelling”) through classically forbidden barrier region. Exponentially sensitive to height and width of barrier.
Repulsive Coulomb interaction
Assume a Boltzmann distribution for the KE,
Probability of nuclei having MeV energy is
Nuclear separation x
Strong nuclear force (attractive)Examples of Tunnelling
Tunnelling 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 energyExamples of Tunnelling
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!
VacuumExamples of Tunnelling
3. 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
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
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)
Asymptotic solution in the limit of very large y:
Equation for H(y):
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)
Label normalizable functions H by the values of n (the quantum number)
Hnis known as the nth Hermite polynomial.
High n state (n=30)
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
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.
Nuclear separation x
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