Just a quick question about some basic terms in QM

[oliver.smith8]

I'm trying to ge my head around some basic things but where ever I look I seem to get a different answer.

what is an eigenvalue, eigenvector, eigenfunction and eigen stat, all in the context of QM?
and how do these apply to the operator equation.

$$\hat{O} \psi = O \psi$$

what is the differnce between a $$\Psi$$ and a $$\psi$$?

I hope somebody can help because I can't get my head round it!

 

[malawi_glenn]

there are many different notations, but one usally denotes operators with hats, and big Psi for a general state-function and small psi's for eigenfunctions (bases).

now look at

$$\hat{O} \psi = O \psi$$

The operator O is acting on small phi, and you get the eigenvalue O, thus small psi is an eigenstate (eigenfuction, eigenvector depending on what psi resembles).

Look at

$$\hat{O} \Psi = \hat{O} ( \psi_1 + \psi _2 + \psi _3) = O_1 \psi _1 O_2 + \psi _2 + O_3 \psi _3$$

The Big Psi is a linear combination of eigenstates to the operator O.

Eigenstate is the most fundamental, eigenvector you have in discrete cases, like spin. And eigenfunctions when you have continuous variables (momentum, position etc).

You will learn all of this by practice, don't panic

If you have done linear algebra, you should recognize these things.

 

[tiny-tim]

Welcome to PF!

Hi oliver! Welcome to PF!

(have a psi: ψ and an Ô )

In Ô = O, Ô is the operator, O is the eigenvalue, and ψ is the eigenvector (same as eigenstate) …

so for example if Ô is the momentum operator, then O is a momentum value, and ψ is a wave function with that (pure) momentum.

I think the difference between Ψ and ψ is just that ψ tends to be used for eigenvectors (eigenstates),while Ψ is used for anything.

 

[dx]

I'm assuming you know what a vector space is.

In quantum mechanics, states are members of a complex vector space $\mathcal{H}$, called Hilbert space. Observables are Hermitian operators on this vector space. An eigenvector of an operator $\hat{O}$ is any vector $v$ that satisfies the equaton $\hat{O} v = \alpha v$, where $\alpha$ is a scalar, called the eigenvalue corresponding to $v$. The eigenvectors of observables (Hermitian operators) form an orthonormal basis set for $\mathcal{H}$, i.e. any state can be written as a linear combination of these eigenvectors. In the context of QM, the eigenvectors are usually called eigenstates. For example, if $\hat{X}$ is the position operator, it's eigenvectors are called position eigenstates.