Let
represent a simultaneous eigenket of the position operators
belonging to the eigenvalues
. Note that, because
the position operators are *fixed* in the Schrödinger picture, we do not
expect the
to evolve in time. The wavefunction of the system
at time
is defined

(267) |

The Hamiltonian of the system is taken to be

The Schrödinger equation of motion, (229), yields

where use has been made of the time independence of the . We adopt the Schrödinger representation in which the momentum conjugate to the position operator is written [see Equation (165)]

Thus,

where use has been made of Equation (169). Here, denotes the gradient operator written in terms of the position eigenvalues. We can also write

where is a scalar function of the position eigenvalues. Combining Equations (268), (269), (271), and (272), we obtain

(273) |

which can also be written

This is Schrödinger's famous wave equation, and is the basis of wave mechanics. Note, however, that the wave equation is just one of many possible representations of quantum mechanics. It just happens to give a type of equation that we know how to solve. In deriving the wave equation, we have chosen to represent the system in terms of the eigenkets of the position operators, instead of those of the momentum operators. We have also fixed the relative phases of the according to the Schrödinger representation, so that Equation (270) is valid. Finally, we have chosen to work in the Schrödinger picture, in which state kets evolve and dynamical variables are fixed, instead of the Heisenberg picture, in which the opposite is true.

Suppose that the ket is an eigenket of the Hamiltonian belonging to the eigenvalue : i.e.,

(275) |

The Schrödinger equation of motion, (229), yields

(276) |

This can be integrated to give

(277) |

Note that only differs from by a phase-factor. The direction of the vector remains fixed in ket space. This suggests that if the system is initially in an eigenstate of the Hamiltonian then it remains in this state for ever, as long as the system is undisturbed. Such a state is called a

(278) |

Substituting the above relation into the Schrödinger wave equation, (274), we obtain

where , and is the energy of the system. This is Schrödinger's time-independent wave equation. A

Such a solution is only possible if

(281) |

Since it is conventional to set the potential at infinity equal to zero, the above relation implies that bound states are equivalent to negative energy states. The boundary condition (280) is sufficient to uniquely specify the solution of Equation (279).

The quantity , defined by

(282) |

is termed the

The Schrödinger time-dependent wave equation, (274), can easily be transformed into a conservation equation for the probability density:

The

We can integrate Equation (284) over all space, using the divergence theorem, and the boundary condition as , to obtain

(286) |

Thus, the Schrödinger wave equation

(287) |

where denotes the expectation value of the momentum evaluated at time . Clearly, the probability current is indirectly related to the particle momentum.

In deriving Equations (284), we have, naturally, assumed that the potential is real. Suppose, however, that the potential has an imaginary component. In this case, Equation (284) generalizes to

(288) |

giving

(289) |

Thus, if then the total probability of observing the particle anywhere in space decreases monotonically with time. Thus, an imaginary potential can be used to account for the disappearance of a particle. Such a potential is often employed to model nuclear reactions in which incident particles are absorbed by nuclei.