For free particle, its frequency and wavelength are not variable with time and position, so it can be described by a plane wave. If a particle is under a variable force field, it can not be described by a plane wave. In any case, we can describe it by a function, namely wave wave function
The phase velocity
dr u dt k
In contrast with electromagnetic wave, matter waves show dispersion in a vacuum. The relativistic energy theorem for free particle
Charpter II Wave Function
2.1 Matter Waves 2.2 The Statistical Interpretation of Matter Waves 2.3 Mean (Expectation) Values in Quantum Mechanics 2.4Three Quantum Mechanical operators 2.5 The Superposition Principle in Quantum Mechanics 2.6 The Heisenberg Uncertainty Principle
2
2
w m0c2 k u ... k k 2m0
w E m c2 c 2 u k k p mv v
We define the group velocity (群速) as
d d ( ) dE vg dk d (k ) dp
The group velocity is identical with the particle velocity
2.1 The Matter Wave
According to De Broglie’s propose, matter (electron, neutron) possesses wave-particle duality. The energy and momentum of matter waves can be expressed:
0 0
0
The shorter the wavelength, the shorter the distance between two point that can be distinctly through the microscope.
2.2 The Statistical Interpretation of matter wave (物质波的统计诠释)
E m c p c
2 2 4 0
2 2
m0 is the rest mass (静止质量) When v<<c,
2 p 2 4 E mc2 m0 c p 2c2 m0c2 ... 2m0
m0 c k (k ) ... 2m0
Phase velocity
According to
p
h

For photon, m0=0,
photon hc / E
In nonrelativistic limit, for electron and neutron, v << c,
p E 2 2m0
2
h / 2m0 E
So we obtain
பைடு நூலகம்
photon (10keV) 1.20 A, electron(1keV) 0.39 A, Neutron (5 eV) 0.13A
Whether this wave should be assigned physical reality?
How to interpret a wave describing a particle?
Yong’s two-slits-diffraction experiment S
P S1 S2 d Q D B
The intensity of wave function at one position is proportional to the probability that particle presents there.
wave function is called probability wave The absolute square of wave function is the probability that particle presents there
E h
h p k

For a free particle, a plane wave is assigned:
(r , t ) A exp[i(k r t )]
The phase of the wave
(r , t )
k r t constant
A
How is the electron twoslits-diffraction experiment?
Two slits synchronously
Open one by one
2.2-1 Probability Wave
• Born’s (波恩) statistical interpretation:
Example
Calculate the wavelength of X ray (with energy 10 KeV ), eletron (1 keV) and neutron (5 eV). Solution: the momentum of particle is
2 2 p ( E / c)2 m0 c