9.1 The Particle in a box Forbidden region Forbidden region 0 a x Infinitely high potential barriers Fig Potential barriers for a particle in a box

Requirements The particle cannot be outside the box. The wavefunction inside box is continuous with the function outside.

To obtain wavefunctions ･･･ describe a particle in the box The particle always has a certain definite energy The de Broglie waves

To obtain wavefunctions ･･･ Relations of physical quantities –energy and momentum

To obtain wavefunctions ･･･ Boundary condition and superposition But neither wavefunction satisfy the boundary conditions (9.2)… We can think the wavefunction which gives the same energy as the wavefunction (9.3). So we think superposition of (9.3) and (9.8)

To obtain wavefunctions ･･･ Determine the unknown constant So (9.9) can be rewritten to (9.11).

To obtain wavefunctions ･･･ Wavenumber So the wavenumber k should take following value. Substituting this result in (9.6), we obtain

To obtain wavefunctions ･･･ Normalization We substitute (9.11) in this.

The wavefunction for a particle in a box As a result of the above arguments, we get the wavefunction. This is associated with a definite energy. Does this also hold for the momentum? This is clearly not the case.

The wavefunction for a particle in a box This describes both a wave with k=nπ/a and a wave k= － nπ/a. We would find values p=hk/2π and p= － hk/2π with equal probabilities. Then, what is the wavefunction which gives a certain momentum?

The probability of measuring a given momentum k. To compare (9.16a) and (9.17)… The component of (9.16a) is a factor of smaller than the corresponding component of (9.17). On the other hand, we expect both components occur with probability=1/2. To go from to 1/2, we square.

Conclusion The wavefunction of the particle in a box which has a certain definite energy is represented as superposition of de Broglie waves. And the energy of particle has the form The probability of measuring a given momentum k can be obtained by taking the square of the absolute value of the coefficient in front of the normalized plane wave.