Chapter (5) Matter Waves

Chapter (5) Matter Waves

De Broglie wavelength

Wave groups Consider a one- dimensional wave propagating in the positive x- direction with a phase speed v p. Where v p is the speed of a point of constant phase on the wave. This wave can be written as x y Acos( ft) (1) Acos( kxt) Take k f : theangular : wavenumber & f v k p frequency f phasevelocity k ()

Superposition of two waves ) cos( ) cos(,, 1 1 1 1 1 1 t x k A t x k A y y y v v A A p p (3) we can write } ) ( ) {( 1 }.cos ) ( ) {( 1 cos ) ( 1 ).cos ( 1 cos cos cos 1 1 1 1 t x k k t x k k A y get we b a b a b a (4) with using

With small different values of 1 &, k 1 & k k=k - k 1, and = - 1 Equation (4) may be interoperated as a board sinusoidal envelope k Acos( x t) Limiting or modulating a high-frequency wave within the envelope 1 1 cos[ ( k k) x [( 1) 1 t ]

Superposition of two waves of slightly different wavelength with t= 0, is given in the following figure.

From (4), the phase velocity v p is (within the envelope) v p v g k ( 1) / ( k k ) / ( 1 ) / ( k k ) / 1 k In the general case, many waves having a continuous distribution of wavelengths must be added to form a packet 1 1 k Here the high frequency wave moves at the phase velocity v 1 of one of the waves or at v because v v 1, also ω 1 ω is small ω 1 ω The group velocity v g (the envelope itself) is 1 v 1 v g d dk k 0 where, k 0, is the central wave number of the many waves present

Limited cases The smaller the spatial width of the pulse, x, the larger the range of wavelengths or wave numbers, k, needed to form the pulse. This may be stated mathematically as If the time duration, t, of the pulse is small, we require a wide spread of frequencies,, to form the group. That is,

A pure sine wave, with extends from - to A superposition of two sin waves of nearly equal wavelength The resultant of the addition of many sine waves (with different wavelengths and possibly different amplitude)

When we have a single sine wave, k was zero (only one k) and x was infinite. As we increase k (by adding more waves) we decreased x (the wave become more confined )

Matter Wave Packets The wave packet and the particle move at the same velocity, According to de Broglie, where E and p are the relativistic energy and momentum of the particle, respectively, The phase speed of these matter waves is given by from then

Now, let us show that the group velocity of the matter wave packet is the same as the particle speed.

Substituting for Using the definition v g, we get mc v p c 1 k d k dk 0 With vp k vp k The phase velocity of matter waves is in other side we have where v is the particle speed Comparing: v g =v,i.e the group velocity of the matter wave packet is the same as the particle speed.

THE DAVISSON GERMER EXPERIMENT(197) Aim: Direct experiment proof that electron has wavelength Three parameters are varying : 1- electron energy; - Nickel target orientation α ; and 3- scattering angle. For constant electron energies of about 100 ev, the scattered intensity rapidly decreased as increased. The intensity of scattered electrons with angle were observed as shown in the figure. h p

Scattering of electrons from a crystalline Ni target leads to electron diffraction

Scattered electrons energy E = 54 ev

Consider the case with α = 90.0, V =54.0 V, and = 50.0, corresponding to the n = 1. To calculate the de Broglie wavelength () for this case: v? h p h mv from

Thus the wavelength of 54.0-V electrons is The experimental wavelength may be obtained by considering the Nickel atoms to be a reflection diffraction grating in excellent agreement with the de Broglie formula

1- We have or k x p x p k k 1becomes p or p x -

- E h f E, E f f t 1 becomes E t or E t

The photon must be scattered through an angle ranging from - θ to + θ, which consequently imparts to the electron an x momentum value ranging from

Thus the uncertainty in the electron s momentum is After passing through the lens, the photon lands somewhere on the screen, but the image and consequently the position of the electron is fuzzy because the photon is diffracted on passing through the lens aperture From optics, the smallest distance x, between two points in an object that will produce separated image in a microscope is given by

then

EXAMPLE (1) The Uncertainty Principle Changes Nothing for Macroscopic Objects (a) Show that the spread of velocities caused by the uncertainty principle does not have measurable consequences for macroscopic objects (objects that are large compared with atoms) by considering a 100-g racquetball confined to a room 15 m on a side. Assume the ball is moving at.0 m/s along the x axis. Solution: Thus the minimum spread in velocity is given by This gives a relative uncertainty of which is certainly not measurable

THE WAVE PARTICLE DUALITY The field (x, y, z, t ) called the wave function represent the matter waves. is a complex number, =a+ib, *=a-ib. It is used to calculate the probability unit volume of finding a particle at a given time in a small of space (x, y, z),which is given by

Is the electron a wave or particle? answer

Experiment shows the impossibility of measuring simultaneously both wave and particle properties, and illustrates the use of the wave function. A parallel beam of electrons falls on a double slit, which has individual openings much smaller than D, a detector is at a distance much greater than D 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.

tan p p y x h p D x