THE SCHROEDINGER EQUATION, WAVEFUNCTIONS AND OPERATORS
AND OPERATORS_
It should be obvious from the previous discussion that we need to be able to evaluate Ч'(г) in all cases. As an example of such a necessity we note that if we have the probability P(r) corresponding to a wavefunction T^r), we automatically have the charge density n(r) corresponding to this vF(r) by the obvious relation
We therefore seek an equation for vF(r). This equation plays a central role in quantum mechanics and is called the Schroedinger equation after the famous Austrian physicist Erwin Schroedinger. The two-slit experiment can be interpreted by a wavefunction vF(r) = (г^+Ч^Гг), where the *F( (75) are spherical waves from the pinholes. Therefore,
'F(r) must be a solution of a wave equation in three dimensions. From previous courses we know that a wave equation must have the form
where к has the dimensions of a wavenumber.
The value of к must conform with the principle of De Broglie, who associated with each particle of momentum p a matter wave of wavelength X
where h is Planck’s constant. Equation 1.4 is usually rewritten in terms of the wavenumber к
with ft(pronounced h bar) = h/2n = Planck’s constant divided by 2n. Given that the energy of any particle can be analyzed into kinetic and potential energy V, i.e.
we deduce that
where m = mass of the particle. Then equation 1.3 takes the form
Equation 1.7 is the time independent Schroedinger equation. We stress that we have not proven or derived anything. All the great equations of physics, like Newton’s 3rd law or Maxwell’s equations of electromagnetism, are ingenious generalizations and abstractions of observations and we have merely followed this course. In fact, we have a generalization: the two-slit experiment and the associated equation 1.3 pertain to electrons in vacuum. We have generalized it (or Schroedinger has in fact done so) to electrons in an arbitrary potential energy V(x,y,z). The justification lies in the results. So far this equation has been used to successfully interpret myriad phenomena with no internal contradictions. A small caution on notation is necessary. In subsequent parts of the book where devices are analyzed we would be required to solve the Poisson equation for the potential which we will denote by V. Then the potential energy of an electron will be equal to (-eV). The reader will be warned of such a change in notation when it occurs.
We now rewrite equation 1.7 in the following form
If we treat
as an operator, equation 1.8 becomes
i.e. an eigenvalue equation where E„ are the eigenvalues and VF„ the eigenvectors. We have now reached a second deeper stage in our development of quantum mechanics. Firstly, the mathematical language of quantum mechanics is linear algebra. Every stationary, time independent, wavefunction 'i'(r) is an eigenstate of the hamiltonian H. Secondly, there is a further generalization. The form of the hamiltonian H is such that it implies a relation between quantum mechanics and classical physics. If we look at this equation we observe that a) it has the dimensions of energy and b) the hamiltonian is derived from the classical expression for the energy (equation 1.5) if we substitute for the momentum p the operator (-itiV), where i = y/-l.
We are now ready to generalize our conceptual framework. This generalization is a big logical jump. It was originally an educated guess of ingenious physicists like Schroedinger and Heisenberg, but their ideas have stood the test of time. The rules of quantum mechanics that we are going to use (and it’s only a subset of the complete set) are the following
i. Write down the classical expression for the energy, this is the classical hamiltonian.
ii. In this expression change all momenta p by the differential operator (—*7?V).
iii. The algebraic expression has now been turned into an operator, the hamiltonian operator. Compute its eigenvalues and eigenfunctions. Then the average value of repeated measurements of the position x of an electron with wavefunction 'Т(г) is (remember l^2 is probability)
and in 3 dimensions (3D)
where in equation 1.11 V stands for volume. The average momentum (written directly in 3 dimensions) is
where in equation 1.12 we have written the grad in bold to make the 3-dimensional character of the equation explicit. We will not repeat this.
Since the square of T'(r) is a probability we must have
i.e. the 'F(r) must be normalizable.
So far we have discussed stationary phenomena—the two-slit experiment is stationary as the fringes do not move with time. The Schroedinger equation discussed up to now is the time independent Schroedinger equation. But there is a time dependent equation. For the stationary systems we have discussed up to now, like that of the two-slit experiment, we have implicitly assumed that the wavefunctions must have a time dependent factor of £ j(
the form e~,ml =e ■ This is in full accordance with both the de Broglie hypothesis and Plank’s principle that E = Tno. Note that this form is exactly the same as that of the stationary waves in electromagnetism and, most importantly, it gives a stationary charge density because
where in 1.14 v|/(r) is the space dependent part of Ч'(гТ).
Differentiating such a non-stationary wavefunction we get
But НЧК = ЕЧ1. Hence
This is the time dependent Schroedinger equation. Although we have derived it for non- stationary wavefunctions with a harmonic variation it holds for all type of time variations. We will not make extensive use of it in this book. We will mainly tackle stationary problems. With the exception of the important section 1.8, the symbol 'P will denote a stationary wavefunction. Before moving to specific problems we note that the hamiltonian operator H belongs to the family of hermitian operators. These operators have real eigenvalues (an obvious requirement) and are defined by Hm n = H*nm where the symbol * denotes complex conjugate.
