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Posts Tagged ‘Quantum Mechanics’

A particle of mass m is confined to a one-dimensional region 0≤x≤a.  At the beginning, the normalized wave function is

Ψ(x,t=0) = √(8/5a)  [ 1 + cos(πx/a)] sin(πx/a).

a) What is the wave function at a later time t=t0?.

b)  What is the average energy of the system at t=0 and t=t0?

c)  Find the probability that the particle is found in the left half of the box (0≤x≤a/2) at t=t0.

Solutions

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Ψ(x,t) is a solution of the Schrödinger equation for a free particle of mass m in one dimensional and Ψ(x,0) =  A exp(-x^2/a^2)

a)  Find the probability amplitude in the momentum space at time t=0.

b)  Find Ψ(x,t).

Solution

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Consider a one dimensional time-independent Schrödinger equation for some arbitrary potential V(x). Prove that if a solution Ψ(x) has the property that Ψ(x) →0 as x→±∞, then the solution must be nondegenerate and therefore real, apart from a possible overall phase factor.

Solution

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In a former post, I described two experiments: the photoelectric effect and the black body radiation. I finish the topic in this post by explaining three more experiments:

3) Frank-Hertz experiment

The experiment of Franck and Hertz consisted of bombarding atoms with monoenergetic electrons and measuring the kinetic energy of the scattered electrons. The amount of the energy absorbed in the collision can thus be estimated. SupposeE0, E1, E2, … are the sequence of quantized energy levels of the atom and and T is the kinetic energy of the incident electron. As long as T is below ∆=(E1– E0), the atoms cannot absorb the energy and all collisions are elastic. As soon as T>(E1– E0), inelastic collisions occur and some atoms go into their first excited states. Similarly, atoms can be excited to second and higher excited states. Thus Frank-Hertz experiment established the quantization of the atomic energy levels.

4) Davisson-Germer experiment

L. de Broglie, seeking to establish the basis of a unified theory of matter and radiation, postulated that matter, as well as light, exhibited both wave and particle aspects. The first diffraction experiments with matter waves were performed with an electron beam by Davisson and Germer (1927). The incident beam was obtained by accelerating  electrons through an electric potential. Knowing the parameters of the crystal lattice, it was possible to deduce an experimental value for the electron wavelength. The result was in agreement with de Broglie  relation, λ=h/p, where h is Planck’s constant and p is the momentum of electron. Similar experiments were later performed using beams of hydrogen molecule and helium atoms. Hence, it showed the wavelike nature of matter.

5) Compton scattering

Compton observed the scattering of X-rays by free or weakly bounded electrons and found the wavelength of the scattered radiation exceeded that of the incident radiation. The difference, ∆λ,  varied as a function of the angle θ between the incident and scattered directions:

where h is the Planck’s constant and m is the rest mass of the electron. Furthermore, ∆λ is independent of the incident wavelength. The Compton effect cannot be explained by any classical theory of light. Hence, it is a confirmation of the photon theory of light.

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In late 19th and early 20 century, a few physical experiments changed our understanding of the nature fundamentally. Later, they led physicists to Quantum mechanics. In this post, I give a brief overview of the most important experiments.

1) Photoelectric effect

This refers to emission of electrons observed when one irradiates a metal under vacuum with ultraviolet light. it was found that the magnitude  of the electric current thus produced is proportional to the intensity of the striking radiation provided that the frequency of the frequency of the light is greater than a minimum value characteristic of the metal, while the speed of electrons does not depends on the light intensity, but on its frequency. These results could not be explained by classical physics.

Einstein in 1905 explained these results by assuming light, in its interaction with matter, consisted of packets of energy ,called photon energy. when a photon encontures an electron of the metal it is entirely absorbed, and the electron, after receiving the energy , spends an amount of work W equal to its binding energy in the metal, and leaves with a kinetic energy of

½ mv2 = hν – W

This quantitative theory of photoelectrons has been completely verified by experiment, thus establishing a quantum nature of light.


2) Black body radiation

A black body is one which absorbs all the radiation falling on it. The spectral distribution of the radiation emitted by a black body can be derived from the general laws of interaction between matter and radiation. The experissions deduced from the classical theory are known as Wien’s law and Rayleight’s law. The former is a good approximation in the short wavelength part of the spectrum, while the latter is in agreement with long wavelength experiments but leads to infinite total energy.

Planck in 1900 succeeded in removing the discrepancy by postulating that energy exchanges between matter and radiation do not take place in a continuous manner but by discrete quantities of energy. He showed that by assuming that the quantum  of energy was proportional to the frequency, ε = hν, the expression of the spectrum, in agreemnet with experiments, is 

where h is a universal constant, known as Planck’s constant. Planck’s hypothesis was confirmed by a set of experiments. Note that in the limit of short and long wavelengths, this equation with reduce to the Wien/Rayleigh forms.

In the next post, I describe the following three expriments:

3) Frank-Hertz experiment

4) Davisson-Germer experiment

5) Compton scattering

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