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The Schrödinger equation is given by
The only simple quantity in this expression is E, which is the energy of a system. H is the Hamiltonian, and Y is the so-called wave function . The wave function is probably the most important concept in quantum mechanics, because it contains all the information of a system. This is rather abstract. So let's have a closer look at the wave function.
A wave function is a function. This means that it has arguments, and to each set of values for the arguments it assigns a value. The arguments of a wave function are generally all coordinates of all particles in a system. So if our system is some molecule, then the arguments are all the x, y, and z coordinates of the nuclei and the electrons. It can be more convenient to use other coordinates. For example, for an atom it is easier to use spherical coordinates for the electrons; i.e., for each electron we use the distance r to the nucleus, and two angles q and j for the orientation of the electron with respect to the nucleus. The wave function for an atom is then written as function of the coordinates X, Y, and Z of the nucleus and all coordinates r, q, and j for the electrons. A wave function is in general a very complicated function. If there are N particles in a system, it is a function with 3N arguments, because that is the number of coordinates of all the particles. If possible, we try to reduce this number of arguments. For an atom or molecule, we are only interested in the wave function for the electrons. This wave function has as arguments only the coordinates of the electrons and not those of the nuclei. (For electrons the wave function also depends on the spin of the electrons. We will ignore this here.)
The information that is easiest to get from the wave function is where the particles of a system are. Quantum mechanics, however, tells us that particles do not have a well-defined position, and we can only talk about probabilities to find particles at specific positions. To illustrate this let's suppose that our system consists of just one particle that can move in just one direction. In this case the wave function Y has just one argument. Let's call this argument x; the coordinate of the particle. The value of the wave function at position x is then Y(x). The probability that we will find the particle between coordinates x en x+dx where dx is an infinitesimal interval is given by |Y(x)|2dx. We see that it is not the wave function itself that gives the probability. That is because a wave function need not be positive. In fact, a wave function can have complex values. To get probabilities (i.e., positive numbers) we first take the absolute value and then square the result.
To get the probability that the particle is between x1 and x2 we have to integrate the probabilities |Y(x)|2dx from x1 to x2. This yields
We can find the probability that the particle is found between -¥ to ¥ by integrating from -¥ to ¥. This probability must, of course, be 1, because the particle must be somewhere. So we have
This is called a normalization condition , and if this relation holds then the wave function is said to be normalized.
The interpretation of the wave function above is called the Copenhagen interpretation , as it was worked out in the years following 1925 by people who were invited by Niels Bohr to Denmark to discuss problems in quantum mechanics. The interpretation can be extended to the properties of the system. Suppose we have a property A that depends on the position x of the particle; i.e., if the particle is at x then the value of the property is A(x). Because the particle has not a well-defined position, we can not talk about the value of that property for the particle. We can talk about the average value áAñ of the property, which is mostly called the expectation value of the property (or, very confusingly and not very precise, just the value of the property). The averaging has to be done with the probabilities |Y(x)|2dx. This means that the expectation value áAñ of A is given by
| áAñ = |
ó
õ |
¥
-¥
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dx|Y(x)|2A(x). |
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This expression can be used for all properties that depend only on the position; e.g., the potential energy of the particle.
There are also properties that do not only depend on the position. An example would be the kinetic energy of the particle. To get the expectation value for such a property we rewrite the last expression as
| áAñ = |
ó
õ |
¥
-¥
|
dx Y(x)*AY(x). |
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The * indicates that a complex conjugate should be taken. (This means if z = a+ib then z* = a-ib where a and b are real numbers and i2 = -1.) Because Y*Y = |Y|2, this doesn't seem to be different from the expression we already had. Note, however, that we have dropped the argument of the property A in the integrand. We have done this because the A on the right-hand-side should be the operator of the property. If the property depends only on the position, then the operator is A(x) and indeed the expression is equivalent to what we already had. If the property is, for example, the kinetic energy, then the operator is -(h2/8p2m)(d2/dx2). Now the new way of writing the expression makes sense, because there has to be a wave function to the right of an operator.
The extension to wave functions that depend on more coordinates should be clear. |Y|2 gives the probability for each coordinate having a specific value. The integrals for the expectation values of properties become multiple integrals over all coordinates.
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