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Example text

41) are called orthogonal functions. 2 Wave functions ø i and probability densities jø i j2 for a particle in a onedimensional box of length a. said to be orthonormal. 3. 6 Tunneling As a second example of the application of the SchroÈdinger equation, we consider the behavior of a particle in the presence of a potential barrier. 3. The region where x , 0 is labeled I, where 0 < x < a is labeled II, and where x . a is labeled III. Suppose a particle of mass m and energy E coming from the left approaches the potential barrier.

5 Particle in a one-dimensional box As an illustration of the application of the time-independent SchroÈdinger equation to a system with a speci®c form for V (x), we consider a particle con®ned to a box with in®nitely high sides. 1. 30) is given by "2 d 2 ø ‡ Iø ˆ Eø À 2m dx 2 for which the solution is simply ø(x) ˆ 0; the probability is zero for ®nding the particle outside the box where the potential is in®nite. 1 The potential energy V (x) for a particle in a one-dimensional box of length a. or d2 ø 4ð2 ˆ À ø dx 2 ë2 where ë is the de Broglie wavelength, 2ð" h ë ˆ p ˆ 2mE p (2X34) (2X35) We have implicitly assumed here that E is not negative.

The solution n ˆ 0 would cause ø(x) to vanish everywhere and is therefore not acceptable. Negative values of n give redundant solutions because sin(Àè) equals Àsin è. We have found that only distinct values for the de Broglie wavelength satisfy the requirement that the wave function represents the motion of the particle. 35) only distinct values En of the energy are allowed ø(a) ˆ A sin n2 ð2 "2 n2 h2 ˆ , n ˆ 1, 2, 3, F F F (2X39) 2ma2 8ma2 so that the energy for a particle in a box is quantized.

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