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Ch 40 Quantum Mechanics © 2005 Pearson Education

Schr Ö dinger Equation K.E. P.E. Total energy © 2005 Pearson Education Give U(x) and ψ(x)  What is E?

Free particle © 2005 Pearson Education

40.1 Particle in a box © 2005 Pearson Education

Normal modes of vibration for a string

Schr Ö dinger Equation of particle in a box Inside a box: With U(x) = 0, © 2005 Pearson Education At x=0, ψ(0) = A 1 + A 2

© 2005 Pearson Education energy levels, particle in a box

Example 40.1 Find the lowest energy level for a particle in a box if the particle is a electron in a box 5x m across, or a little bigger than an atom. Find the lowest energy level for a particle in a box if the particle is a electron in a box 5x m across, or a little bigger than an atom.ANS: © 2005 Pearson Education

Is proportional to the probability finding the particles

normalization condition particle in a box © 2005 Pearson Education

40.2 Potential Wells © 2005 Pearson Education U(x)=0

© 2005 Pearson Education Square-well potential Inside the well: Where U=0: Where U=U 0 : outside the well:

© 2005 Pearson Education Wave function

© 2005 Pearson Education Probability distribution

40.3 Potential Barriers and Tunneling © 2005 Pearson Education Cannot pass through Can pass through

© 2005 Pearson Education Potential- energy barrier

© 2005 Pearson Education Tunneling

40.4 The harmonic Oscillator © 2005 Pearson Education

For S.H.M

© 2005 Pearson Education

energy levels, harmonic oscillator © 2005 Pearson Education

40.5 Three-Dimensional Problems three-dimensional Schrödinger equation © 2005 Pearson Education

To be a solution of the Schrodinger equation, the wave function ψ (x) and its derivative dψ(x)/dx must be continuous everywhere, except where the potential- energy function U(x) has an infinite discontinuity. Wave functions are usually normalized so that the total probability for finding the particle somewhere is unity.

The energy levels for a particle of mass m in a box (an infinitely deep square potential well) with width L are given by Eq. (40.9). The corresponding normalized wave functions of the particle are given by Eq. (40.13). (See Examples 40.1 and 40.2) © 2005 Pearson Education

In a potential well with finite depth U 0, the energy levels are lower than those for an infinitely deep well with the same width, and the number of energy levels corresponding to bound states is finite. The levels are obtained by matching wave functions at the well walls to satisfy the continuity of ψ(x) and d ψ(x)/dx. (See Examples 40.3 and 40.4) © 2005 Pearson Education

There is a certain probability that a particle will penetrate a potential energy barrier although its initial kinetic energy is less than the barrier height. This process is called tunneling. (See Example 40.5)

The energy levels for the harmonic oscillator, for which U(x) = 1/2k’x 2,are given by Eq. (40.26). The spacing between any two adjacent levels is Ћω, where is the oscillation angular frequency of the corresponding Newtonian harmonic oscillator. (See Example 40.6) © 2005 Pearson Education

The Schrodinger equation for three-dimensional problems is given by Eq. (40.29). © 2005 Pearson Education

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