Quantum Theory

Read e-book online Basic Quantum Mechanics PDF

By Klaus Ziock

ISBN-10: 0471983829

ISBN-13: 9780471983828

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2a). The ball rolls up the slope, and a short time later at x = 0 (position 2) all the kinetic energy is transformed into potential energy V(O) = E. At this moment the ball reverses its direction and starts to roll back down the slope. Never under any circumstances will it run up the hill any further; in other words, never will it be found in the region x > 0, where Vex) > E. Now we investigate what happens under similar circumstances in a quantum mechanical system. We assume that a particle with kinetic energy E approaches a potential barrier of height Vo > E.

To simplify Eq. 44) For convenience, we shall drop the primes henceforth and write x instead, keeping in mind that our unit of length is now: ~ li/mOJ. Innocent though it looks, Eq. 44 is not easy to solve. 46) This is usually called the asymptotic solution of Eq. 44. 47) x2 /2J(x) into Eq. 44, hoping that this will simplify our problem. 48 ) 6 We can easily convince ourselves that this has, indeed, the dimension of a length. 4. 40 SOME SIMPLE PROBLEMS Substituting Eqs. 48 into Eq. f(x) = f(x) + 2xf'(x) or f"(x) - 2xf'(x) - f"(x) - x2f(x) + f(x)(J..

We notice that the ratio of the coefficients is the same as that given by Eq. 56. 5 so that the highest power of x is always positive. This amounts to a choice of the phase of the wave function which has no physical significance. The constants Bn can be obtained from the normalization. 70) The normalization as defined in the example (Eq. 71) 10 For a derivation of this expression, see J. L. Powell and B. , Reading, Mass. 1961. 30 0. , ~ c. 20 £ 0. 45 Displacement from equilibrium position Displacement from equilibrium position (b) (bi Fig.

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Basic Quantum Mechanics by Klaus Ziock


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