By John C Morrison
Significant e-book for enthusiastic undergraduate scholars to get the fundamentals of Quantum idea from a phenomenological point of view
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Rather than following Schrödinger’s original line of argument, we will show that the Schrödinger equation arises naturally in the context of the ideas of de Broglie and the results of electron diffraction experiments. 1 THE WAVE EQUATION The electron diffraction experiments of Davisson and Germer and Thompson and Reid conﬁrmed de Broglie’s suggestion that a wave is associated with the motion of a particle. According to eqs. 1) and p = k. 2) A relation between the kinetic energy of the particle and the angular wave vector can be obtained by writing the kinetic energy in terms of the momentum as follows: KE = Modern Physics for Scientists and Engineers Copyright © 2010 by Elsevier, Inc.
12) the Schrödinger equation becomes d2 ψ + k2 ψ = 0. 1 The Wave Equation and ψ(x) = B sin(kx) are solutions of eq. 13). For instance, substituting ψ(x) = A cos(kx) into eq. 13) gives d2 A cos(kx) dx 2 + k2 A cos(kx) = −Ak2 cos(kx) + k2 A cos(kx) = 0. The cosine function is an even function of x being unchanged when x is replaced by −x, and the sine function is an odd function of x changing its sign when x is replaced by −x. We consider ﬁrst the even solutions. Inside the well, the even solutions have the form ψ(x) = A cos kx, for − L/2 ≤ x ≤ L/2.
The work function W is the energy required to free the least tightly bound electrons from the metal. More energy will have to be supplied to more tightly bound electrons to free them from the metal. For a particular frequency of light, the maximum kinetic energy of the emitted electrons is equal to the energy of the photons given by eq. 1) minus the work function of the metal. We thus have (KE)max = hf − W. 3) The value of Planck’s constant h and the work function W of the metal can be calculated by plotting the maximum kinetic energy of the emitted electrons versus the frequency of the light as shown in Fig.