# Two-fermion system wave function

• PhyPsy
In summary: In this case, the antisymmetric space function is not the best wave function to use, because it leads to too much overlap of the electrons. In contrast, the symmetric space function leads to a smaller mean separation of the electrons, which is what we want in this case.
PhyPsy
For a two-electron atom, this book says that the overall wave function is either a) the symmetric space function times the antisymmetric spin function or b) the antisymmetric space function times the symmetric spin function. However, in another problem which involves two fermions in a harmonic oscillator, it says the overall wave function is a sum of the two aforementioned products (a + b). Why would the atom's wave function not be a + b instead of being just a or b?

If your two-electron atom does not have a superposition of different energy levels for the electrons (something you have to carefully prepare), the spins are fixed and antisymmetric OR symmetric, but not both, and the same is true for the spatial component. If you have superpositions of different energy levels, you might get something like "a+b".

For the first excited state of the atom, one electron is in the lowest energy level n=1, and the other is in the level n=2. According to your statement, there should be a superposition, but the answer only contains a b (from my initial post) term and no a term.

PhyPsy said:
For the first excited state of the atom, one electron is in the lowest energy level n=1, and the other is in the level n=2.
If you fix the energy levels like this, you do not have a superposition of different energy states.

What I meant is something like Rabi oscillations, where the electrons are not in specific energy levels (at least not all the time).

OK, so then for systems that are not a superposition, how do you know whether to use a or b? I understand that for atoms where both electrons have the same value for n, the antisymmetric space function is equal to 0, so you have to pick a, but what about atoms where the electrons have different values for n? Is it always b for atoms with electrons that have different values for n?

In the case of electrons, usually the antisymmetric spatial wave function is preferrable, since this leads to a larger mean separation of the electrons (look up the Hund rules).

But both are valid wave functions, and which one is preferable depends on the circumstances. For example, most molecules have singlet ground states, but then there is oxygen2, which does not.

## 1. What is a two-fermion system wave function?

A two-fermion system wave function is a mathematical function that describes the quantum state of a system that consists of two identical fermions (particles with half-integer spin) in a specific spatial configuration. It takes into account the position, momentum, and spin of each fermion and describes the probability of finding the system in a particular state.

## 2. How is the two-fermion system wave function different from the single-fermion wave function?

The two-fermion system wave function is different from the single-fermion wave function in that it takes into account the interactions between the two fermions, while the single-fermion wave function only describes the state of one particle. Additionally, the two-fermion system wave function must be antisymmetric, meaning it changes sign when the positions of the two fermions are exchanged, in order to satisfy the Pauli exclusion principle.

## 3. What is the significance of the two-fermion system wave function in quantum mechanics?

The two-fermion system wave function is significant in quantum mechanics because it allows us to understand and predict the behavior of systems that consist of two identical fermions, such as atoms with two or more electrons. It also demonstrates the fundamental principles of quantum mechanics, such as the Pauli exclusion principle and the concept of particle exchange symmetry.

## 4. How is the two-fermion system wave function calculated?

The two-fermion system wave function is calculated using the Schrödinger equation, which is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. It involves solving a complex mathematical equation that takes into account the interactions between the two fermions and their respective quantum states.

## 5. Are there any real-life applications of the two-fermion system wave function?

Yes, there are many real-life applications of the two-fermion system wave function. It is used in the field of quantum chemistry to predict and study the behavior of molecules with multiple electrons. It is also used in condensed matter physics to understand the properties of materials with multiple fermions, such as metals. Additionally, it has applications in nuclear and particle physics, as well as in the development of quantum technologies.

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