|
The bracket notation was developed by Paul Dirac (Nobel prize 1933) to make it easier to write the equations of quantum mechanics. We will not discuss the full power of the notation, which can make many complicated derivations a matter of simple manipulations of the symbols in the equations, but we will see how it drastically reduces the amount of writing of integrals.
Let's start with one electron and ignore its spin. We then have the following bracket notation for an integral.
| áY|Fñ = |
ó
õ |
dr Y(r)*F(r). |
|
Here Y and F are two (arbitrary) wave functions for the electron, and the integration is over the whole space where the electron can be found. If we include the spin we get
| áY|Fñ = |
ó
õ |
dr ds Y(r,s)*F(r,s), |
|
where s is the formal variable that we associate with spin. Note that the bracket notation leaves out the variables of the integration. You have to know what the variables are of the wave function to know what kind of integration the bracket stands for. Spatial and spin variables are often combined in the single variable t. The last equation then becomes
| áY|Fñ = |
ó
õ |
dt Y(t)*F(t). |
|
The name of the notation refers to the symbols á and ñ which are (angular) brackets. The parts of áY|Fñ are called bra (áY|) and ket (|Fñ). The whole is also called an inner product, because it has all the properties of an inner product.
If we have two electrons with coordinates and spin t1 and t2, we get
| áY|Fñ = |
ó
õ |
dt1 dt2 Y(t1,t2)*F(t1,t2), |
|
with the wave functions now, of course, being functions of all spatial and spin coordinates of both electrons. Again, you have to know what the coordinates of the wave functions are to know what integral the inner product stands for. The extension to more electrons should new be clear. If we have N electrons then
| áY|Fñ = |
ó
õ |
dt1 ... dtN Y(t1, ... tN)*F(t1, ... tN), |
|
is the meaning of áY|Fñ.
There is another type of integral that we often encounter. For one electron we have
| áY|A|Fñ = |
ó
õ |
dt Y(t)* AF(t), |
|
where A is an operator working on the wave function F in the ket. The left-hand-side is the bracket notation for the integral. It is also called a matrix element. (In calculations operators are generally replaced by matrices. The matrix elements of these matrices are given by integrals as in the expression above.) Again áY| is called the bra and |Fñ the ket. For N electrons we get
| áY|A|Fñ = |
ó
õ |
dt1 ... dtN Y(t1, ... tN)*F(t1, ... tN). |
|
|
|
