Understanding Dirac Notation in Quantum Mechanics

[phyky]

1.) an inner product of a state vector represent by <\psi|\psi>. sometimes the notation is like <\phi|\psi> is mean transfer from state |\psi> to <\phi|.it mean the former 1 do not transfer the state? what is the difference between both?
2.) what is mean by <x|\psi>? is it mean x(position) mean the final state we reach where we extract form |\psi> ? from eigen value equation \hat{x} |\psi>=\lambda|\psi> where \lambda is the eigen value of position. so why do we need <x|\psi>?

 

[DrClaude]

1.) an inner product of a state vector represent by <\psi|\psi>. sometimes the notation is like <\phi|\psi> is mean transfer from state |\psi> to <\phi|.it mean the former 1 do not transfer the state? what is the difference between both?

It is similar to the dot product of ordinary vectors. It is not a "transfer", but rather a projection. $\langle \psi | \psi \rangle$ is the "length" of the state $\psi$. It is equal to 1 if the state is normalized.

2.) what is mean by <x|\psi>? is it mean x(position) mean the final state we reach where we extract form |\psi> ?

I will not go into the fact that $\langle x |$ is not a proper bra, but basically writing
$\langle x | \psi \rangle$ allows to given a representation of the state $| \psi \rangle$ as a wave function of the cartesian coordinate $x$,
$$
\psi(x) = \langle x | \psi \rangle
$$

from eigen value equation \hat{x} |\psi>=\lambda|\psi> where \lambda is the eigen value of position. so why do we need <x|\psi>?

$| \psi \rangle$ is an arbitrary quantum state, not an eigenvector of the $\hat{x}$ operator.

 

[phyky]

is projection kinda same as the ψ "collapse" to 1 of the eigenstate \phi?

 

[K^2]

No. Do you know anything about dot-products from linear/vector algebra? The <|> inner products are just generalization of dot-product. Phi and psi are vectors, and \langle \phi | \psi \rangle is just a dot-product between the two. Of course, in the continuous case, the sum over components becomes an integral over position. Hence if you are given \langle x | \psi \rangle = \psi(x) and \langle x | \phi \rangle = \phi(x), then \langle \phi | \psi \rangle = \int \phi^*(x)\psi(x) dx.

 

[AlbertEi]

But if $\phi$ is an eigenfunction of $\psi$, then:
\begin{equation}
\langle \phi | \psi \rangle
\end{equation}
is the transition amplitude, right? (I think that is maybe what the OP is asking.)

 

[K^2]

But if $\phi$ is an eigenfunction of $\psi$,

|\phi\rangle is a vector. So is |\psi\rangle. So one can't be an eigen vector of the other. Because they are both vectors. To have an eigen vector, you must be dealing with a map from a Hilbert space to Hilbert space. Such a map is called an operator. So if A is an operator, then for every |\psi\rangle there exists a corresponding vector |A\psi\rangle such that |A\psi\rangle = A|\psi\rangle. Furthermore, if |\psi\rangle is an eigen vector of A, then A|\psi\rangle = \lambda |\psi\rangle for some \lambda.

 

[AlbertEi]

Yeah, you are completely right. Sorry. I meant if |\phi\rangle is a basisvector of the Hilbert space, then my statement is correct?