Exchange Interaction - Helium

In summary, the Hamiltonian of a Helium atom is represented by H = H_1 + H_2 + W, which includes the energies of the two electrons and their interaction. To find the energy of the system, the expectation value is calculated by taking the inner product of the ket and bra states. In this case, a linear combination of the two possible state functions is used for both the ket and the bra.
  • #1
Master J
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Looking thru some notes one this, I came across something which I can't get my head around.

The Hamiltonian of the system (Helium atom) is H = H_1 + H_2 + W, the energies (kinetic and electron-nucleus) of electron 1, electron 2 and the interaction between the electrons themselves.

To find energy, one does the usual expectation value, whereby the H acts on a ket state, and this is then acted on by a bra state. What gets me here tho is that the explanation uses

1) for the ket, a linear combo of the 2 possible statefunctions, ie. either electron could be considered to be in a particular place.

2) for the bra, he first uses only one state, and then does the whole derivation over again with the other.

I am asking, is this simply the same as uses the linear combo for the bra also?
 
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  • #2
It makes sense to me that you'd need 2 state functions (ket & bra) to get an expectation value, yet I'm not sure this is the same situation?Yes, this is the same as using a linear combination of the two possible state functions for both the ket and the bra. The expectation value of a Hamiltonian is the average energy of a system when it is in a particular state. The expectation value is calculated by taking the inner product of the ket and bra states. This means that both the ket and the bra must be expressed in terms of the same basis set so that the inner product can be calculated correctly. Thus, in this case, a linear combination of the two possible state functions is used for both the ket and the bra.
 

1. What is exchange interaction in helium?

Exchange interaction in helium is a quantum mechanical phenomenon that describes the repulsion between two electrons with the same spin. It is responsible for the stability of helium atoms and plays a crucial role in the properties of helium gas.

2. How does exchange interaction affect the properties of helium gas?

Exchange interaction between electrons in helium gas results in a strong correlation between their positions and spins. This leads to the observed properties of helium gas, such as its low boiling point, high thermal conductivity, and non-reactivity.

3. Can exchange interaction be observed in other elements?

Yes, exchange interaction is present in all elements with more than one electron. However, its effects are most noticeable in elements with low atomic numbers, such as helium, because of the simplicity of their atomic structures.

4. How is exchange interaction related to the Pauli exclusion principle?

The Pauli exclusion principle states that no two electrons in an atom can have the same set of quantum numbers. Exchange interaction arises from this principle, as electrons with the same spin must be in different quantum states to avoid violating it.

5. Is exchange interaction a purely repulsive force?

No, exchange interaction can also have attractive effects. In certain cases, such as in the formation of chemical bonds, exchange interaction can lead to a net attractive force between electrons with opposite spins. This can contribute to the stability of molecules.

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