The Born interpretation means that many wavefunctions which would be acceptable mathematical solutions of the Schrodinger equation are not acceptable because of their implications for the physical properties of the system.
For example, the wavefunction must not be infinite over any finite region. If it is, then the integral of the square modulus of the wavefunction is equal to infinity, and the normalisation constant is zero. Unfortunately, this implies that the particle described by such a wavefunction has a zero probability of being anywhere where the wavefunction is not infinite, but is certain to be found at all points where the wavefunction is infinite.
| It is clearly absurd to suggest that the particle can be definitely located at multiple positions, so a wavefunction such as this is deemed an unacceptable solution: |
Note that this does not rule out the possibility of a wavefunction rising to infinity at a single point. (Mathematically speaking, the area under such a curve at the point where it is infinite is given by an infinite height multiplied by an infinitely narrow width, the value of which may be finite.)
Again the normalisation constant would be zero, and the particle has a zero probability of being located at positions where the wavefunction is non-infinite. It is certain to be found at the point where the wavefunction is infinite, but this is acceptable. This is the wavefunction of a particle that is precisely located at one definite point in space.
The Born interpretation also renders unacceptable solutions of the Schrodinger equation for which |ψ|2 has more than one value at any point. This would suggest that there were multiple different probabilities of finding the particle at that point, which is clearly absurd.
| The requirement that the square modulus of the wavefunction must be single-valued usually implies that the wavefunction itself must be single valued, and this is the requirement that we shall normally impose. This is an example of a wavefunction that violates this requirement.
The grey lines indicate the region where the wavefunction is multivalued. |
The Born interpretation also renders unacceptable solutions of the Schrodinger equation for which |ψ|2 has more than one value at any point. This would suggest that there were multiple different probabilities of finding the particle at that point, which is clearly absurd.
Further restrictions arise because the wavefunction must satisfy the Schrodinger equation, which is a second-order differential equation. This implies that the second derivative of the function must exist. This in turn implies that the first derivative of the wavefunction must exist (as if it does not then the second derivative is also undefined and the wavefunction cannot be a solution of the Schrodinger equation).
| The first derivative of a function gives its gradient at a given point, and it thus exists as long as the function is continuous – only if there is a break in the function is there a point at which its first derivative does not exist. This is thus an unacceptable wavefunction. | |
| Further, the requirement that the second derivative of the wavefunction should exist means that wavefunctions of this form are also not acceptable:
The second derivative of this wavefunction is discontinuous at the point indicated, where the gradient of the line changes by more than 180º. In practice, this requirement may be somewhat flexible, particularly if the potential energy of the system shows rapid changes with distance. |