Schrödinger Wave Equation

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

(3.48) |

The Hamiltonian of the system is taken to be

The Schrödinger equation of motion, (3.10), 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 (2.74).] Thus,

where use has been made of Equation (2.78). 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 (3.49), (3.50), (3.52), and (3.53), we obtain

(3.54) |

which can also be written

This is the

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

(3.56) |

The Schrödinger equation of motion, (3.10), yields

(3.57) |

This can be integrated to give

(3.58) |

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

(3.59) |

Substituting the previous relation into the Schrödinger time-dependent wave equation, (3.55), we obtain

where , and is the energy of the system. This is the

Such a solution is only possible if

(3.62) |

Because it is conventional to set the potential at infinity equal to zero, the previous relation implies that bound states are equivalent to negative energy states [50]. The boundary condition (3.61) is sufficient to uniquely specify the solution of Equation (3.60).

The quantity , defined by

(3.63) |

is termed the

The Schrödinger time-dependent wave equation, (3.55), can easily be transformed into a conservation equation for the probability density:

The

We can integrate Equation (3.65) over all space, using the divergence theorem [92], and the boundary condition as , to obtain

(3.67) |

Thus, the Schrödinger time-dependent wave equation conserves probability. In particular, if the wavefunction starts off properly normalized, according to Equation (3.64), then it remains properly normalized at all subsequent times. It is easily demonstrated that

(3.68) |

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 (3.65), we have, naturally, assumed that the potential is real. Suppose, however, that the potential has an imaginary component. In this case, Equation (3.65) generalizes to

(3.69) |

giving

(3.70) |

(See Exercise 2.) Thus, if then the total probability of observing the particle anywhere in space decreases monotonically with time. Hence, an imaginary potential can be used to account for the disappearance or decay of a particle. Such a potential is often employed to model nuclear reactions in which incident particles are absorbed by nuclei.