It is much wiser to express the operators in spherical coordinates, so we can use them any time we need them in a problem that is best described in this coordinate system.
This can be done using the chain rule, as we saw in previous chapters. We will now consider the total energy that is, the sum of the kinetic energy plus the potential energy.
However, in contrast to the kinetic energy term, the potential energy depends on the forces experienced by the particle, and therefore we cannot write a generic expression. If you took a physics course, you may be familiar with different expressions for the potential energy of different systems e.
In all cases, the potential energy depends on the coordinates of the particles. They are quite different from algebraic operators which work on mathematical functions.
You can see why the same word is used in both cases, but you should keep in mind that the two kinds of operators are different. Both kinds of operators are used in quantum mechanics and often in similar kinds of equations, as you will see a little later.
When you are first learning the subject it is well to keep the distinction always in mind. Later on, when you are more familiar with the subject, you will find that it is less important to keep any sharp distinction between the two kinds of operators.
You will, indeed, find that most books generally use the same notation for both! But first, one special remark. Is there some meaning to the complex conjugate of this amplitude?
Many important operators of quantum mechanics have the special property that when you take the Hermitian adjoint, you get the same operator back. So far we have reminded you mainly of what you already know. Now we would like to discuss a new question. How would you find the average energy of a system—say, an atom? Since the system does not have a definite energy, one measurement would give one energy, the same measurement on another atom in the same state would give a different energy, and so on.
What would you get for the average of a whole series of energy measurements? But you may get a different number for each measurement. How are these probabilities related to the mean value of a whole sequence of energy measurements? We continue for, say, a thousand measurements. When we are finished we add all the energies and divide by one thousand. We are almost there.
What we mean by the probability of something happening is just the number of times we expect it to happen divided by the total number of tries. A simple result. Our new formula for the average energy is not only pretty. Then all the arguments go through in the same way. We are asking for the special case of Eq. You can begin to see how we can go back and forth from the state-vector ideas to the wave-function ideas.
The quantity in the braces of Eq. We should make one qualification on our results. Notice the similarity in form between Eq. There are, however, operators for which this is not true. For them you must work with the basic equations in You can easily extend the derivation to three dimensions. With Eq. All we need is the wave function. An operator is a generalization of the concept of a function applied to a function.
Whereas a function is a rule for turning one number into another, an operator is a rule for turning one function into another. Such an equation, where the operator, operating on a function, produces a constant times the function, is called an eigenvalue equation. The function is called an eigenfunction, and the resulting numerical value is called the eigenvalue. Eigen here is the German word meaning self or own. We will discuss this in detail in later Sections.
Add a comment. Active Oldest Votes. Or in other words, to the question: If the state of a physical system is completely determined in terms of definite outcomes of some observable s , are there still measurements that can be made that are not definitively determined? Once one gets used to thinking of observables as forming an algebra, some questions naturally appear: Does our algebra actually get all the physical observables, does everything we can measure correspond to something in the algebra of classical observables?
If not, do physical observables require loosening the rules of this algebra somewhat, e. Improve this answer. Stan Liou Stan Liou 7, 1 1 gold badge 20 20 silver badges 34 34 bronze badges. The question amounts to why the language is linear algebra. Featured on Meta. Now live: A fully responsive profile. Linked 1.
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