Central potential. The hydrogen atom
6.6 The structure of atoms
6.7.3 The Pauli exclusion principle
If we try to write the two complete wave function of the system as the product of two single electron wave functions as in (6.7.2) we run into a problem. The two wave functions may not describe the same single electron state (ϕ1(r1) 6= ϕ2(r2)), therefore we must determine which electron is in which state.
So the question arises: can we distinguish between the two electrons, can we identify them?
Example 6.4. Please note that we did not ask howwe would do this, we asked whether it is possible at all. This is a very subtle difference, which has an enormous effect. A simple thought experiment may illustrate it. Let us consider a box which contains two objects.
These objects (e.g. gas molecules, electrons) can move around in the box randomly. Let us further suppose that we are unable to distinguish between these objects. Divide the
box into two equal partitions. Now determine the probability of finding one object in each partition!
Refer to the two partitions of the box as “left” and “right” respectively. If we assume the objects are distinguishable – in principle, (but maybe not in practice) – then we can label them, let us say with A and B. There are 4 different possibilities of the distribution of the objects in the boxes, each with the same probability as seen in the next figure:
Therefore the probability of one object in both partitions is 1/4 + 1/4 = 1/2. So if we measure this probability as 1/2, we can conclude that the objects in the box are, in fact distinguishable.
Now let the object be indistinguishable even in principle. In this case we cannot label the objects and the following figure shows the different possibilities.
Therefore the probability of one object in both partitions is 1/3. So if the measured probability is 1/3, we can conclude that the objects in the box we cannot distinguish are, in fact indistinguishable.
Important 6.7.1. Experiments show that electrons are indistinguishable objects in re-gions where their wave functions overlap.
This means that we can distinguish between two electrons when they are e.g. in two separate free atoms, but we cannot distinguish between electrons of the same atom.
This we can take into account by writing the wave function as a linear combination of all possible ordering of the one-electron wave functions
ϕ(r1,r2)± = 1
√2(ϕ1(r1)ϕ2(r2)±ϕ1(r2)ϕ2(r1)) (6.7.4) where the 1/√
2 factor is the normalization constant. We can only measure the absolute square of the wave function and for both signs it is the same
|ϕ+(r1,r2)|2 =|ϕ−(r2,r1)|2 (6.7.5)
The combination ϕ+ is asymmetric function for the exchange of the coordinates of the two electrons
ϕ+(r1,r2) =ϕ+(r2,r1), while
ϕ−(r1,r2) = −ϕ−(r2,r1)
is antisymmetric. It is easy to see from (6.7.4) that the antisymmetric wave function is ϕ− ≈0, when the two electrons are in the same atom and described by the samen, `, m quantum numbers and r1 ≈r2. This means that the probability density of two electron atoms with antisymmetric (spatial) wave functions is higher when the two electrons with the same quantum numbers are apart from each other, i.e. this is the preferred state.
If the electrons are further apart the effective shielding of the nucleus will be smaller, the other electron will bound to the nucleus tighter with lower energy. For symmetric (spatial) wave functions no such restriction apply. Therefore the energy of the electrons with antisymmetric wave functions is smaller then that for symmetric wave functions.
(Note that these energy levels can be measured spectroscopically.)
The complete wave function of the electron in an atom however includes the spin as well. The spin of the two electrons can be parallel (s = s1 +s2 = 1) or anti-parallel (s =s1−s2 = 0). In the first case the z-component can be ms =−1,0,1, which is called a triplet and has symmetric spin functions, while in the second case ms = 0, which is a singlet that is an anti-symmetric spin function. If the function for “up” spin of the first electron is χ1,↑and for “down” spin isχ1,↓ and for the second the index 2 is used then the complete spin function can be one of the following combinations
χs= The complete wave function also can be symmetric or antisymmetric:
ϕs(r1,r2) = Spectroscopic measurements show that the symmetric spatial ϕ+ state is always a sin-glet, which requires the antisymmetric spin function χa, while ϕ+ is always a triplet, i.e.
it must have the symmetric spin function χs.
It follows the complete electron wave functionmust always be antisymmetric.
This principle is true not only in an atom and not only for electrons, but for any systems consisting of the same half-integer spin particles (e.g. electrons, protons or neutrons).
Important 6.7.2. The complete wave function, which includes the spin, of a system of any half-integer spin particles must be antisymmetric for the exchange of the coordinates of any two particles.
This is called the Pauli exclusion principle or simply the exclusion principle.
The name exclusion principleemphasizes the consequence of the antisymmetric nature of the complete wave function, – and not only in an atom – namely that no two electrons can be in the same state, where all of their quantum numbers would be equal.
This is another, equivalent phrasing of the Pauli exclusion principle.
In an atom the exclusion principle says that no two electron wave functions may have all of their four quantum number to be the same. But if we consider two non interacting free atoms, than it is possible that they have electrons with the samen, `, m, msquantum numbers.
Important 6.7.3. The Pauli exclusion principle results in a strong repulsive force be-tween electrons of a multi-electron system that does not allow them to occupy the same state. This force is electrostatic in nature, therefore the exclusion principle does not introduce any new kind of force.
The wave function of an N-electron system (e.g. a multi-electron atom) depends on the quantum numbers and coordinates of all of the electrons. To make the equations easier to read we denote a given combinations of the four quantum numbers n, `, m, ms with a single letter and replace all arguments with a number, i.e.
{n, `, m} ⇒ a {n0, `0, m0} ⇒ b
ϕa,b,...(r1,r2, . . .)⇒ ϕa,b,...(1,2, . . .)
It is easy to see that with this notation the antisymmetric spatial wave function of a two electron system can be written as a determinant:
√1
Similarly for N electrons (variables)
ϕabc... = 1
√N!
ϕa(1) ϕb(1) ϕc(1)· · · ϕa(2) ϕb(2) ϕc(2)· · · ϕa(3) ϕb(3) ϕc(3)· · ·
... ... .... ..
(6.7.9)