Wave properties of matter 8.2 Wave Nature of Matter De Broglie Wavelength Diffraction of electrons Uncertainty Principle Wave Function Tunneling Material particles behave as waves with a wavelength given by the De Broglie wavelength (Planck’s constant/momentum) l=
h p
The particles are diffracted by passing through an aperture in a similar manner as light waves. The wave properties of particles mean that when you confine it in a small space its momentum (and kinetic energy) must increase. (uncertainty principle) This is responsible for the size of the atom. De Broglie Wavelength properties of particle waves. Momentum of a photon ­ inverse to wavelength. p =
E c
Einstein’s special relativity theory E=
since h l
p=
Lois De Boglie hc l
Wavelength of a particle­ inverse to momentum. l=
h p
De Broglie proposed that this wavelength applied to material particles as well as for photons. (1924) Big particles­ small wavelengths. Find the De Broglie wavelength of a 100 kg man walking at 1 m/s. Suppose you walk into a room through a doorway. In the wave picture you will be diffracted. By a small amount since you are big. But suppose you can shrink in size. Then the angle will increase. Small particles­large wavelengths Find the wavelength of an electron traveling at 1.0 m/s (m e =9.11x10 ­31 kg) p=mv l=
h
h
6.63x10 -34 Js =
=
= 6.6x10 -36 m p mv (100kg)(1.0m / s)
For macroscopic momenta the wavelengths are so small that diffraction effects are negligible. l=
h
6.63x10 -34 =
= 7.3x10-4 m mv (9.11x10-31 )(1)
= 0.73 mm Diffraction effects should be observable for small particles. The wavelength of the electron can be changed by varying it velocity.
1 Diffraction of light from crystals Diffraction of electrons from crystals Davisson­Germer Experiment (1927) An electron beam is scattered from a crystal Ni crystal Crystals act as a three­dimensional diffraction grating Light with wavelength close to the inter­atomic spacing (x­rays) is diffracted. Comparison between electron diffraction and x­ray diffraction George Thompson (1927) The scattered beam shows a diffraction pattern expected for the crystal spacing. Electron microscopy Diffraction pattern Aluminum powder Either x­rays or electrons The electron wavelength was adjusted to the same value as the x­ray by varying the voltage. The diffraction pattern for x­rays and electrons are very similar. X­ray pattern electron pattern Material particles have wave properties. Wavelength in an electron microscope l=
h
mv
Shorter wavelength can be obtained by increasing v, the speed of the electron. Electron microscopy Suppose an electron microscope accelerated electrons across a potential of 10 4 V. What would the wavelength of the electron be? KE =
1 2 mv = eDV 2
v =
l=
2eDV m e h
h
m
1 =
= h mv m 2eDV
2me DV
l = 6.63x10-34
­ + DV e­
1 = 1.22x10 -11 m 2(9.11x10-31 )(1.60x10-19 )(10 4 )
electron microscope pictures of ribosomes model of ribosome Electron microscopy can be used to measure molecular structure.
2 Diffraction of particles Probabilistic Interpretation of the wave amplitude. Intensity µ Amplitude 2 of the diffracted wave. particle with wavelength
l
But each particle hits a certain point on the screen The amplitude 2 is interpreted as the probability of the particle hitting the screen at a certain position This is true for electrons as well as photons. Wave property of particles
Wavefunction In quantum mechanics the result of an experiment is given in terms of a wavefunction Y. The square of the wavefunction Y 2 is the probability of the particle being at a certain position. The wavefunction can be calculated using using the Schrödinger Equation. For instance for electrons in an atom. Uncertainty Principle
Particle passing through a slit ­­ The uncertainty in position is Dx
The uncertainty in the x component of momentum is x direction Dpx = mDv x The particle is diffracted Dx
x direction Dv x Dx sin q = l =
Dx l
Dv x sin q = q =
v Therefore What accounts for the size of the hydrogen atom? k e 2 electron proton PE = - o r
­ + PE­> ­ infinity as r ­> zero Classical electrostatics predicts that the potential energy of the hydrogen atom should go to – infinity The finite size of the atom is a quantum mechanical effect. DxDp x = h
most often written as an inequality
When Dx decreases, Dp x increases. The size of an atom Dv x v
Dv h Dx x =
v
mv
l
Decreasing the slit reduces Δx But increases the width of the diffraction, Δv x h mv
The position and velocity cannot be know with unlimited certainty. DxDp x > h As the size of the atom r decreases – the kinetic energy increases due to the uncertainty principle. Use linear momentum as a rough estimate.
­ + r =Δx h
h =
Dx r
h Dp x =
» p x p x cannot be smaller than Δp x 2r
Dp x »
KE =
1 p 2 h 2 mv x 2 = x » 2 2
2m 8r m
E= KE+PE goes through a minimum as a function of r KE
E E
PE r 3 Tunneling across a barrier a macroscopic object impinging on a barrier the object cannot penetrate within the barrier. Electron Tunneling Microscope Wall potential energy barrier wave function decays e ­ 0 .01 2 Electron _ Pr obability µ Y » Yo 2 e -2 a r α ~10 nm ­1 distance The probability of tunneling decreases exponentially with the width of the barrier. The tunneling probability falls off exponentially with r and is a sensitive function of distance ele ctron proaba bility a wave particle impinging on a barrier can penetrate within the barrier for distance. and go through the barrier if it is thin enough. 1 E ­07 1 E ­12 1 E ­17 1 E ­22 1 E ­27 0 0 .5 1 1.5 2 2.5 3 r (nm ) Tunneling in Photosynthesis hf e ­ 10 ­12 s 10 ­12 s 2.0 nm
10 ­1 s 10 ­10 s Bacterial Reaction Center Electron transfer time vs distance The electron tunnels through the space between groups. Electron tunneling times are a sensitive function of distance. The back tunneling is 10 11 times slower than the first step. 4