Chapter 5: Matter waves

Inadequacies of Bohr theory:

• It failed to predict the observed intensities of spectral lines.
• It had only limited success in predicting emission and absorption wavelengths for multielectron atoms.
• It failed to provide an equation of motion governing the time development of atomic systems starting from some initial state.
• It overemphasized the particle nature of matter and could not explain the newly discovered wave–particle duality of light.
• It did not supply a general scheme for “quantizing” other systems, especially those without periodic motion.

De Broglie in 1923, postulated that because “==photons have wave and particle characteristics, perhaps all forms of matter have wave as well as particle properties”==.

According to de Broglie, every electron was a wave that guided the the electron through space.

The wavelength and frequency of these matter waves were given by,

De Broglie’s explanation of Quantization

• only a discrete set of wavelengths is allowed for standing waves, while other wavelengths not included in this discrete set rapidly vanish by destructive interference.

• corresponding to a distribution of initial electron velocities, most wavelengths and velocities rapidly die off.

• The residual standing wave patterns thus account for the identical nature of all atoms of a given

  element.

• Assumption: The allowed Bohr orbits arise because the electron matter waves interfere constructively when an integral number ofwavelengths exactly fits into the circumference of a circular orbit.

Davisson–Germer experiment

• Clear-cut proof of the wave nature of electrons was obtained in 1927 by the work of Davisson and Germer.
• The first case was an accidental discovery, and the second involved the discovery of the particle properties of the electron and the wave properties.
• It was an attempt to understand the arrangement of atoms on the surface of a nickel sample by elastically scattering a beam of low-speed electrons from a polycrystalline nickel target.
• Even though, these experiments were not conducted at first as a test of de Broglie’s wave theory, it was concluded as that.
• while the diffraction lines from low-energy reflected electrons are quite broad, the lines from high-energy electrons transmitted through metal foils are quite sharp.
• This effect occurs because hundreds of atomic planes are penetrated by high-energy electrons, because if 2d sinx is not exactly equal to n(lamda), there will be no diffracted wave.
• This sharp bands are because of scattering and cancellation of these waves by destructive interference.
• If de Broglie’s postulate is true for all matter, then any object of mass m has wavelike properties and the de Broglie wavelength.

Electron microscope

• ideas of such a device were tossed about in the cafés and bars of Paris and Berlin as early as 1928.
• electron tubes and circuits, vacuum technology, and electron beam control—all pioneered in the development of the cathode ray tube (CRT).
• The best optical microscopes using ultraviolet light have a magnification of about 2000 and can resolve two objects separated by 100 nm, but a TEM using electrons accelerated through 100 kV has a magnification of as much as 1,000,000 and a maximum resolution of 0.2 nm.
• A second type of electron microscope with less resolution and magnification than the TEM, but capable of producing striking three-dimensional images, is the scanning electron microscope (SEM).
• These secondary electrons are detected by a plastic scintillator coupled to a photomultiplier, amplified, and used to modulate the brightness of a simultaneously rastered display CRT.
• 

Wave groups and dispersion

• The matter wave representing a moving particle must reflect the fact that the particle has a large probability of being found in a small region of space only at a specific time.

• all observed waves are limited to definite regions of space and are called pulses, wave groups, or ^^wave packets in the case of matter waves.^^

• A wave group consists of a superposition of waves with different wavelengths, with the amplitude and phase of ^^each component wave adjusted so that the waves interfere constructively over a small region of space.^^

• Outside of this region the combination of waves produces a net amplitude that ^^approaches zero rapidly as a result of destructive interference^^

• \

  

Hiesenberg’s uncertainty principle

• This theory overcame some of the problems with the Bohr theory of the atom, such as the postulate of “unobservable” electron orbits.
• it is impossible to determine simultaneously with unlimited precision the position and momentum of a particle.
• 

Dual nature

• At a distance from the slits much greater than D is an electron detector capable of detecting individual electrons.
• It is important to note that the detector always registers discrete particles localized in space and time.
• In all cases if the detector collects electrons at different positions for a long enough time, a typical wave interference pattern for the counts per minute or probability of arrival of electrons is found
• although the electrons are detected as particles at a localized spot at some instant of time, the probability of arrival at that spot is determined by finding the intensity of two interfering matter waves.
• the interference pattern has been lost and the accumulated result is simply the sum of the individual results. psi1+ps2=psi
• a wave packet localized in a region x contains a range of wavenumbers del k, where del x del k >=1/2 . Because px= hk, ^^this implies that there is an uncertainty principle for position and momentum.^^

In a similar fashion one can show that an energy–time uncertainty relation exists, given by

• In quantum mechanics matter waves are represented by a wavefunction psi(x, y, z, t).
• The probability of finding a particle represented by psi in a small volume centered at (x, y, z) at time t is ^^proportional to psi^2 .^^
• The wave–particle duality of electrons may be seen by considering the passage of electrons through two narrow slits and their arrival at a viewing screen.
• We find that although the electrons are detected as particles at a localized spot on the screen, the probability of arrival at that spot is determined by finding the ^^intensity of two interfering matter waves.^^