# Two-Level System Consideration: Electron Indistinguishability

• aaaa202
In summary: So the state of two particles can be more restricted than the state of one particle.In summary, for a two level system such as the spin 1/2 system, there are infinitely many possible states for a single electron, but only one possible state for two electrons due to the symmetrization requirement. This does not restrict the possible spin orientations for the individual particles, but it is peculiar that the composite system has fewer degrees of freedom than the individual subsystems. This is a result of the indistinguishability of electrons at a fundamental level.

#### aaaa202

Consider a two level system, for example the spin 1/2 system. For a single electron there are infinitely many possible states for this system since any state:
ls> = a lup> + bldown> with lal^2 + lbl^2 = 1
is an allowed state.
Now if we again consider the same system but with two electrons in it, it seems there is not the same freedom for how the state of the system can look. Because by the symmetrization requirement we have:
ls> = 1/sqrt(2)(lup,1>ldown,2)-lup,2>ldown,1>)
And that is the only possible state that meets this requirement. Is this true? If so, I guess it just seems weird to me that introducing a second electron takes away the freedom for the single electron. And how does this change physically happen. Imagine we have an electron in the spin up state and bring it close to another electron. The symmetrization requirement now means there is a 50% chance that the first electron is instead in the spin-down state. I think the answer to the last question is that I'm thinking this too classically and that electrons are indistinguishable on a very fundamental level. But it is nevertheless still weird to me.

The spin is a magnetic moment - i.e. a magnet.
When you put one magnet on a table, you can orient it however you like - but put two on the table and they will assume a limited range of relationships.

aaaa202 said:
Consider a two level system, for example the spin 1/2 system. For a single electron there are infinitely many possible states for this system since any state:
ls> = a lup> + bldown> with lal^2 + lbl^2 = 1
is an allowed state.
Now if we again consider the same system but with two electrons in it, it seems there is not the same freedom for how the state of the system can look. Because by the symmetrization requirement we have:
ls> = 1/sqrt(2)(lup,1>ldown,2)-lup,2>ldown,1>)
And that is the only possible state that meets this requirement. Is this true?

Yes, but that's more general than it looks. Consider a different basis,

$|U\rangle = a |up\rangle + b |down\rangle$
$|D\rangle = -b^* |up\rangle + a^* |down\rangle$

with $|a|^2 + |b|^2 = 1$

Then we could just as well form $\frac{q}{\sqrt{2}}(|U\rangle |D\rangle - |D\rangle |U\rangle)$
but that happens to be equal to $\frac{q}{\sqrt{2}}(|up\rangle |down\rangle - |down\rangle |up \rangle)$ (because if you multiply it out, the other two possibilities cancel). So this single state doesn't preclude finding the component particles to have spin-up in any arbitrary direction.

But you're right, that there is something a little peculiar about this. Normally, you expect that the number of degrees of freedom for a composite system has to be greater than or equal to the number of degrees of freedom of each subsystem, but that's not the case, quantum mechanically.

## 1. What is a two-level system?

A two-level system is a quantum mechanical system with only two possible energy states. This can refer to a physical system with two energy levels, such as a spin-1/2 particle, or a theoretical model used to describe a more complex system with only two relevant states.

## 2. Why is electron indistinguishability important in two-level systems?

In quantum mechanics, electrons are considered indistinguishable particles, meaning they cannot be identified or tracked individually. In two-level systems, this means that the two energy states cannot be differentiated, and the system must be described as a whole rather than as two distinct parts.

## 3. How does electron indistinguishability affect the behavior of two-level systems?

Electron indistinguishability leads to phenomena such as quantum superposition, where the system can exist in a combination of both energy states simultaneously. It also affects the probabilities of measurement outcomes and the dynamics of the system over time.

## 4. Can two-level systems be used in practical applications?

Yes, two-level systems have been studied extensively in quantum computing, as they can be used as quantum bits (qubits) for information processing. They have also been used in other applications such as in quantum sensors and quantum cryptography.

## 5. How are two-level systems different from multi-level systems?

The main difference between two-level systems and multi-level systems is the number of energy states they have. Two-level systems have only two possible energy states, while multi-level systems have more than two. This leads to different behaviors and applications for these types of systems.