As I understand it, the fact electrons are indistinguishable arises from the probabilistic nature of quantum mechanics, thus the mathematical considerations for a system with distinguishable or indistinguishable particles differ.
Let us imagine an atom that has four electrons. Let us call each electron as Alex, Bill, Charley and Dick. Let us call each electronic orbital as Worker's desk, Manager's desk, Director's desk and President's desk. Then I would like to ask. Who is sitting on which desk? Are they sitting on each own desk forever? Or are they exchanging their own desk with other's desk busily? Please someone describe the state in this atomic company!
Each electron is in a defined quantum state, i. e. electron Alex is in the worker's desk, and so on. You can only change quantum states by an energy conversion process (for example, light absorption or emission). However, electrons are indistinguishable particles, meaning you can't name one electron Alex, the other one Bill, and so forth. That is because the uncertainty principle prevents us from determining the exact position of an electron inside the electron cloud, therefore you can't distinguish one from another through a set of coordinates. When you solve the Schrodinger equation to obtain the energies of the quantum states, you need to account for that indistinguishability.
You wrote that each electron is in a defined quantum state.
A defined quantum state has its own eigen energy. Right?
Therefore the total energy of the four electrons might be simply the sum, i.e.,
Etotal=E1+E2 +E3+E4 eq (1)
Is this correct?
And you suggested that when we solve the Schroedinger equation to obtain the energies of the quantum states, we need to account for the indistinguishability.
Is the above equation (1) still correct? Can we really calculate the compornent eigen energies E1,...,E4, separately?
Right, each electron is in a defined quantum state and each one has its own energy. The time-independent Schrodinger equation states HY = EY, where H is the hamiltonian operator, Y is the wave function and E is the energy.This means that Y is a en eigenfunction of the hamiltonian operator. Let's first analyze the case of the hydrogen atom (1 single electron and a nucleus). If you assume that the nucleus is static relative to the electron, due to the huge difference in mass, then the hamiltonian operator consists of the kinetic energy of the electron plus the electromagnetic potential energy between the electron and the nucleus. The hydrogen atom is the only case in which you can solve the schrodinger equation, mathematically. When you consider polyelectronic atoms, for example helium, the interlectronic repulsion energy appears in the hamiltonian operator:
Etotal = K1 + K2 + V1 + V2 + I1-2, where K is kinetic energy, V is potential energy and I is interelectronic repulsion energy.
This prevents us from finding a mathematically exact solution. To solve this problem, approximation methods are used.
Now, the indistinguishability of the electrons must be accounted for in the wave function. For polyelectronic atoms, you used hydrogen-like wave functions, Slater orbitals, etc. It's a bit more complicated because you also need to take into consideration the spin and the Pauli exclusion principle.
Thank you for your detailed and kind explanation on atomic physics.
But it seems that you missed one point of my question posted in the previous page.
Yes, the time-independent Schroedinger equation is,
If we represent the position of each electron j by (xj, yj, zj), then the wave function Y may be a function of all the electronic positions, that is,
And the eigenvalue E should be a single scalar value. I would repeat E is a single scalar value.
My question is this.
Can we resolve the total eigenvalue E into the sum of compornent values like E1,E2, ... each of which represents the each electron's energy?
I think that the situation in an atom might rather be different.
At least in principle, we might become able to solve the many variable Schroedinger equation (2) analytically someday in future. On that time, we can calculate the entire, total eigenvalue E, not the individual "orbital's" or "electron's" energies E1, E2,... Thus we could know the total sum of energy of all the electrons involved in an atom, but we cannot know the individual electron's energies.
And this situation seems to be just the case of "quatum mechancal entanglement"[ref. "A quantum threat to Special relativity" Scientific American March 2009, page 26]. That is, we do not know the individual energies, but we know only the total energy.
How do you like this kind of discussion? Am I wrong?
P.S. Please feel free to continue your discussion of how to take into consideration the spin and Pauli's exclusion principle. Please show me.
Regarding the issue about the total energy E, truly we can separate in components E1, E2, E3,... but as I said before, those energies would only represent the kinetic and potential energies of their respective electrons. To obtain an accurate value (in accordance with experimental measurements) we must also take into account inter-electronic repulsion energy. The problem is this energy depends on the position of all the electrons at once, thus it can't be separated in many components.
The issue about calculating the total energy instead of particular energies is really interesting. I am no expert in quantum physics though, but certainly science is always open for new developments that ultimately lead to a better understanding of the universe.
About the Pauli exclusion principle, it states that the total wave function must be asymmetrical with respect to the exchange of electrons. This like even and odd functions (even * even = even, even * odd = odd, and so on). A wave function can be separated in its spatial and spin components. For example, for helium in its fundamental state, we could write the following wave function using hydrogen-like orbitals:
Y = (1/2)^0.5 1s(1)1s(2)[a(1)B(2) - B(1)a(2)]
The first number is the normalization factor, the second one is the spatial component, and the last one is the spin component. In this example, the spin equation is anti-symmetric with respect to the exchange of electrons, meaning that we can't define which electron has up or down spin.
When trying to write wave functions for higher atomic number elements, it is far more easy to use Slater determinants.
I don't know if this was helpful to you, but I would recommend the following book: Physical Chemistry A Molecular approach by Donald McQuarrie. It is a really good book to understand all this.
Thank you a lot for your response.
Attached a file, please open it first for continuing the discussion.
Yse, the problematic term is the inter-electronic repulsion energy eq(7). Exactly correct! But does the method of Slater determinant wave fucntion take into consideration this inter-electronic term eq(7) correctly?
Do we know how accurately the Slater determinant wave function eq(8) satisfies the original Schroedinger equation in which the Hamiltonian operator may be represented by eq(6)? Do we know? How accurate is the Slater determinant wave function? The original idea seems to have started from the Hartree method in which an average filed was assumed to be valid approximation. Then Hartree constructed a set of associated integro-differential equations (when I was young, I myself too was entranced with the idea.)
But do you believe in the idea of "average field" concept which is generated from the other electron's distribution? If an electron is at a position r, then the total electron density distribution constituted from the other electrons needs to be completely zero at that position r. Can the spatial charge distribution that generates the average field satisfy this requirement?
Are the concepts of atomic orbital or the molecular orbital really valid and reliable? How is your opinion? What is your textbook saying about this point?
Thank you again for participating in this dicussion.
Notation.docx 22.5 K