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By Darwin C. G.
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Additional info for A Quantum Theory of Optical Dispersion
E .. .. .. x .. x0 .. (a) 47 .. uE (x).... .. .. .. .. ... .. .. . .. . .. . . . ........... .. . x .. x0 .. 1: Potential energy and wavefunction for E < V (x) case. 2 Bound states This is the case where E > V (x) only in some regions of ¯nite x and E < V (x) for jxj ! 1. States satisfying these conditions are called bound states because the corresponding classical case is that of a particle restricted to a certain region of space.
We shall now derive the forms of the position and momentum operators and their eigenstates in the position representation just de¯ned. We shall do this in one space dimension. Extension to three dimensions is straightforward (problem 1). e. 5) For an arbitrary state jsi the result of operation by X is Xjsi. Its position representation is hxjXjsi. 6) This shows that in the position representation the e®ect of operating by X is just multiplication of the wavefunction by the corresponding eigenvalue x.
For the same values of n, show by direct integration that the un are mutually orthogonal. 60. 6. Show that noninteger eigenvalues are not possible for the number operator N. ] CHAPTER 4. SOME SIMPLE EXAMPLES 44 7. 85. 8. For the Landau level problem ¯nd the raising and lowering operators in terms of momentum and position operators. 85 in this case and solve for wavefunctions of the four lowest levels. Chapter 5 More One Dimensional Examples The examples of quantum systems presented in chapter 4 gave some hint as to what to expect in quantum mechanics.