Wave Functions

Another approach was to derive the many particle wave function (where the denotes the coordinates of the 1st electron, the 2nd electron, and so on, and t is time) and solve the stationary (time-independent) Schrödinger equation for the system:

(where is the hamiltonian, i.e., the operator of the total energy for the system), and calculate the set of possible wave functions (called eigenfunctions) and corresponding energies (eigenvalues) . The eigenfunctions have to be physically acceptable, and for finite systems:

1.

they should be continuous functions,

2.

they should be at least doubly differentiable,

3.

its square should be integrable,

4.

they should vanish at infinity (for finite systems),

The eigenfunction corresponding to the lowest energy , describes the ground state of the system, and the higher energy values correspond to excited states. Physicists like to call 's the states (since they contain all possible information about the state), while chemists usually call them wave functions.

Once the function (or its approximation, i.e., in case when the Schrödinger equation is solved only approximately) is known, the corresponding energy of the system can be calculated as an expectation value of the hamiltonian , as:

where denotes the complex conjugate of since in general these functions may produce complex numbers. This is needed since the operator in this case represents a physical observable, and the result has to be a real number. This equation is frequently written using Dirac bra ( ) and ket ( ) notation to save space (and to confuse the innocent):

And if , i.e., the wave function is normalized, the equation looks even simpler:

 
Figure 1: Volume element for a particle

Once we know the wave function for a given state of our system, we can calculate the expectation value for any quantity for which we can write down the operator. The wave function itself does not correspond to any physical quantity, but its square represents the probablity density. In other words:

or

or

represents the probablity that electron 1 is in the volume element around point , electron 2 is in the volume element of the size around point , and so on. If describes the system containing only a single electron, the simply represents the probability of finding an electron in the volume element of a size centered around point . If you use cartesian coordinates, then and the volume element would be a brick (rectangular parallelipiped) with dimensions whose vertex closes to the origine of coordinate system is located at (x, y, z). Now, if we integrate the function over all the space for all the variables (i.e., sum up the probablilities in all the elements ), we should get a probability of finding our electrons anywhere in the Universe, i.e., 1. This is why it is a good idea to normalize fundtion . If it is not normalized, it can easily be done by multiplying it by a normalization constant:

Since square of represents the probablility density of finding electrons, one may suspect, that it should be easy to calculate the total electron density from it. And actually it is:

where N is the total number of electrons, and is the famous Dirac delta function. In cartesians, it simply amounts to integrating over all electron positions vectors but one. Which one, is not important, since electrons are indistinguishable, and a proper wave function has to reflect this:

It is interesting to note, that for the wave function which describes the system containing only a single electron (but only then !!!):

i.e., logically, the electron density and the probability density of finding the single electron are the same thing.

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