lecture10

An example of correspondence principle: large quantum number
Classically, the particle is equally like to be anywhere in the box: Pcl 1
L
Quantum probability density distribution is correspondence principle
L
I
0
II
x
Quantum Mechanics
Example 5.6
Electrons with energies of 1.0eV and 2.0eV are incident on a barrier 10.0eV high and 0.50nm wide. Find their respective transmission probabilities. 1.1×10-7; 2.4×10-7 How are these affected if the barries is double in width? 1.3×10-14; 5.1×10-14
Quantum Mechanics
A particle without the energy to pass over a potential barrier may still tunnel through it
5.10 Tunnel Effect
energy
E
U
Approximate transmission probability III
Quantum Mechanics
Scanning tunneling microscope designer
Shared The Nobel Prize in physics 1986 for their design of the STM
With Ernst Ruska for his Gerd Binnig Heinrich Rohrer fundamental work in electron optics, and for the 19331947design of the first electron Federal Republic Switzerland microscope
dimensionless SS SE well-behaved wave functions require
Quantum Mechanics
Quoted in Robert J. Scully, The Demon and the Quantum (2007), 22.
Quantum Mechanics
Brief Review
Measurements on quantum states Steady-state Schrö dinger equation is the physical law governing the motion of a particle in timeindependent potentials Combined with boundary conditions and restrictions on the wave functions, it predicts the existence of quantized energy levels Mathematically, it is nothing but a specific example for the broad type of eigen problems
x
Quantum Mechanics
Wave functions and probability densities
Quantum Mechanics
Summary for finite potential well
Conclusions for the particle in a box also apply for the finite potential well except for the following differences: Energy for the latter is lower than the counterpart in the particle in a box. Number of bound states depends on the potential depth and width is [x] represents the integer which is no less than x.
What really matters for me is … the more active role of the observer in quantum physics … According to quantum physics the observer has indeed a new relation to the physical events around him in comparison with the classical observer, who is merely a spectator. ---Wolfgang Pauli Letter to Niels Bohr (1955)
Quantum Mechanics
Example 5.5
Find the expectation value <x> of the position of a particle trapped in a box L wide. L/2, the average position of the particle is the middle of the box in all quantum states. Confliction with the fact that probability density is 0 at L/2? The momentum expectation value? <p>=0!?
Quantum Mechanics
Summary for particle in a box
Energy discrete values (quantized): Characteristic of bound state problems in QM, where a particle is localized in a finite region of space. The discrete energy states are associated with an integer quantum number n. Energy of the lowest state (ground state) comes close to bounds set by the Uncertainty Principle. The number of nodes for wave function is n-1. Recover classical probability distribution at high energy by spatial averaging.
Quantum Mechanics
QM modifications anticipated
The allowed energies will not form a continuous spectrum but instead a discrete spectrum of certain specific values only; The lowest allowed energy will not be E=0 but will be some definite minimum E=E0; There will be a certain probability that the particle can penetrate the potential well it is in and go beyond the limits of –A and +A.
of Germany
Quantum Mechanics
STM and atomic force microscope (AFM)
Quantum Mechanics
Summary for tunnel effect
Unbound/scattering state problem. Probability density flux conserved. Quantum tunnel effect: E U , T 0 , can’t be explained classically. In appropriate approximation
Quantum Mechanics
How boundary conditions and normalization determine wave functions

U
5.8 Particle in a Box
Vanishes!
0
L
x
Quantum Mechanics
energy levels (eigenvalues)
Quantum Mechanics
Momentum
momentum eigenvalues of trapped particle momentum eigenvalue eq. corresponding momentum eigenfunctions eigenvalues
Quantum Mechanics
Quantum Mechanics
It’s energy levels are evenly spaced steady state SE
Harmonic oscillator (QM)
introducing the dimensionless quantities scaled config. scaled energy
Quantum Mechanics
5.11 Harmonic Oscillator
energy
1 2 U kx 2
Hook’s law classical eq. of E motion of h.o.
A
0
A
x
amplitude
init. phase
frequency of h.o.
potential energy
quantum numbers
Particle in a box
Well-behaved! Normalized?
Quantum Mechanics
Particle in a box
Normalized eigenfunctions (of Hamiltonian)
Quantum Mechanics
Quantum Mechanics
The wave function penetrates the walls, which lowers the energy levels energy
U
5.9 Finite Potential Well
In region I and III: III
L
I
0
பைடு நூலகம்
E II
Probability amplitudes and densities
# of nodes: n-1 excited states
ground state
Quantum Mechanics
Example 5.4
Find the probability that a particle trapped in a box L wide can be found between 0.45L and 0.55L for the ground and first excited states. 19.8 percent for the ground state 0.65 percent for the first excited state