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Lecture_08: Outline
Matter Waves
 de Broglie hypothesis
 Experimental verifications
 Wave functions
de Broglie hypothesis
Photons:
E ph  h
p  k  h 
Wave → Particle
What about the opposite?
Particle → Wave ?
 h p
de Broglie hypothesis
Matter waves:
h
h
h
B  

p mv
2mK
Wavelength for a walking man?
Wavelength for a moving electron?
What is the wavelengths difference for 5 eV electron
and 5 eV photon?
How to reveal the wave properties of electrons?
Experimental verifications
X-rays diffraction:
X-rays are electromagnetic waves with
λ = 10-8 to 10-12 m = 10 – 0.001 nm
Experimental verifications
Bragg’s law:
2d sin   n
If θ and λ are known,
d can be determined
Experimental verifications
Davisson-Germer experiment:
• Electrons were directed
onto nickel crystals
• Accelerating voltage is
used to control electron
energy: E = |e|V
• The scattering angle
and intensity (electron
current) are detected
– φ is the scattering angle
Experimental verifications
Electron scattering:
From X-ray
experiments:
d = 0.091 nm
2d sin   n
For φ = 50°
(θ = 65°):
λ = 0.165 nm
Experimental verifications
Experimental results:
For φ ~ 50°, the
B 
maximum is at ~54V
E = 54 eV = 8.64·10-18J
h
 0.167nm
2mE
Experimental verifications
X-rays and electrons:
Experimental verifications
Electrons:
Experimental verifications
Other particles:
He atoms
Neutrons
C60 molecules
Experimental verifications
Double-slit experiments:
Light:
Experimental verifications
Double-slit experiments:
Light:
Electrons:
Experimental verifications
Individual electrons:
• In previous experiments many electrons were
diffracted (or show interference)
• Will one get the same result for a single electron?
• Such experiments were performed
– Intensity of the electron beam was so low that only one
electron at a time proceeds
– Still diffraction (and interference) patterns, and not
diffused scattering, were observed, confirming that
Thus individual electrons possess wave properties!!!
Wave functions
Complimentarity Principle:
The particle and the wave models are
COMPLIMENTARY
No measurements can simultaneously
reveal the particle and the wave
properties of matter
Wave functions
Photons:
 

E(r , t )  Emax cos(k  r  t )
E ph  h
p  k  h 
Free Particles:
B  h p


p  E
 (r , t )  A cos(  r  t )


Wave functions
Superposition principle:
Electromagnetic waves:



E(r , t )  E1 (r , t )  E2 (r , t )
Matter waves:



(r , t )  1 (r , t )  2 (r , t )
Wave functions
Probability:
Matter waves:
The value of |Ψ(x,t)|2 gives the probability to
find the particle in point x at time t.

 dx ( x, t )

2
1