Stationary States

(C.63) |

According to Equation (C.20), such solutions have definite energies, . For this reason, they are usually written

The probability of finding the particle between and at time is

(C.65) |

This probability is time independent. For this reason, states whose wavefunctions are of the form (C.64) are known as

This equation is called the

(C.67) |

where is the Hamiltonian. (See Section C.5.)

Consider a particle trapped in a one-dimensional square potential well, of infinite depth, which is such that

(C.68) |

The particle is excluded from the region or , so in this region (i.e., there is zero probability of finding the particle outside the well). Within the well, a particle of definite energy has a stationary wavefunction, , that satisfies

The boundary conditions are

This follows because in the region or , and must be continuous [because a discontinuous wavefunction would generate a singular term (i.e., the term involving ) in the time-independent Schrödinger equation, (C.66), that could not be balanced, even by an infinite potential].

Let us search for solutions to Equation (C.69) of the form

where is a constant. It follows that

The solution (C.71) automatically satisfies the boundary condition . The second boundary condition, , leads to a quantization of the wavenumber: that is,

where et cetera. (A ``quantized'' quantity is one that can only take certain discrete values.) Here, the integer is known as a

(C.74) |

Thus, the allowed wavefunctions for a particle trapped in a one-dimensional square potential well of infinite depth are

where is a positive integer, and a constant. We cannot have , because, in this case, we obtain a null wavefunction: that is, , everywhere. Furthermore, if takes a negative integer value then it generates exactly the same wavefunction as the corresponding positive integer value (assuming ).

The constant , appearing in the previous wavefunction, can be determined from the constraint that the wavefunction be properly normalized. For the case under consideration, the normalization condition (C.32) reduces to

(C.76) |

It follows from Equation (C.75) that . Hence, the properly normalized version of the wavefunction (C.75) is

(C.77) |

At first sight, it seems rather strange that the lowest possible energy for a particle trapped in a one-dimensional potential well is not zero, as would be the case in classical mechanics, but rather . In fact, as explained in the following, this residual energy is a direct consequence of Heisenberg's uncertainty principle. A particle trapped in a one-dimensional well of width is likely to be found anywhere inside the well. Thus, the uncertainty in the particle's position is . It follows from the uncertainty principle, (C.60), that

(C.78) |

In other words, the particle cannot have zero momentum. In fact, the particle's momentum must be at least . However, for a free particle, . Hence, the residual energy associated with the particle's residual momentum is

(C.79) |

This type of residual energy, which often occurs in quantum mechanical systems, and has no equivalent in classical mechanics, is called