Download Atomic and Molecular Physics for Physicists Ben-Gurion University of the Negev

Ben-Gurion University of the Negev
Atomic
Atomic and
and Molecular
Molecular Physics
Physics
for
for Physicists
Physicists
Ron Folman
Chapter 5:
Wave mechanics in quantum mechanics:
The wave function, the uncertainty principle, statistical nature
Main References: Corney A. Atomic and Laser
Spectroscopy, Oxford UP, 1987; QC 688.C67
1987; Chapters in Modern Physics, Open
University
Exercises:
Dudi Moravchik.
www.bgu.ac.il/atomchip
www.bgu.ac.il/nanofabrication
www.bgu.ac.il/nanocenter
Wave mechanics: The mathematical formalism I.
(the wave packet, the uncertainty principle, statistical nature)
Vg=dw/dk
If particles are waves, how do we use infinite
waves to described localized entities?
The wave packet has a finite width and can therefore
be used to describe particles.
How do we make wave packets from waves?
We used many frequencies! Ψ(x)=∫dk Ψ(k) eikx
Fourier
Transform
X
– group velocity ↑- phase velocity
The uncertainty principle ∆x ∆p = h
A “naive” way of getting it:
∆x ⋍ nλ nh/p
∆λ λ/n
∆λ = ∆p h/p2 = h/p ∆p/p = λ ∆p/p => ∆λ/λ = ∆p/p 1/n
∆p p/n
• a bullet (50g) and an electron are both shot with a speed of 300m/s. This velocity
Is measured with an accuracy of 0.01%. With what accuracy can we measure their
Position?
Graphical example of why this happens:
The Bohr/Heisenberg microscope
The apparatus
The electron must have the transverse momentum
uncertainty as the photon (because of momentum
conservation). As the photons that we see come from
an angle of θ the photon transverse momentum
uncertainty is: ∆px = p sinθ = sinθ h/λ
Every microscope has the limit (the so-called
diffraction limit) of observing a point like particle
with a width of ∆x = λ / sinθ . This is then the accuracy
With which we know the particles position
Exercise:
To fight the idea of W2+P2=1, Einstein suggested to place an infinitely sensitive
“which path” detector in one of the two slits of the double-slit experiment
Based on the uncertainty principle, and the fact that the distance between the two
slits should be of order λ (why?), prove that Einstein’s idea cannot work.
Exercise: explain diffraction with the FT, and show that you know how to
estimate the width of the zero order from the uncertainty principle.
Another example of the uncertainty principle: ∆E ∆t > h
Example I: the line width of a pulse laser.
What is the minimal line width of a Femto second laser?
Example II: the line width of an atomic energy level.
Why is the line width of the Rubidium excited level 5MHz?
• Prove another manifestation of the uncertainty principle: namely, that
∆N ∆Φ > h, where N is the number of particles (FOCK state) and Φ is the phase.
The statistical nature of QM: the interpretation of the wave function.
The statistical nature
of the wave function.
Prob.=|ψ|2
where |ψ| is called the amplitude
Think of what a two particle
wave function means?
Ψ(x1,y1,x2,y2,t)