Representing |psi> with Continuous Eigenvectors: An Example

[jinksys]

Homework Statement

for discrete basis vectors {{e_n}}, a state vector |psi> is represented by a column vector, with elements being psi_n = <e_n|psi>. When basis vectors correspond to those with continuous eigenvalues, vectors are represented by functions. Give such an example of a state vector |psi> being represented in a space spanned by position eigenvectors {|x>}.

Homework Equations

The Attempt at a Solution

C_n = <x_n | psi>

I'm really taking a shot in the dark with this one. I'm using Griffith's, so I assume my solutions lies somewhere in chapter 3.

 

[dextercioby]

So...$|\psi\rangle$ is a ket which is part of the space spanned by eigenvectors of X, the latter being kets satisfying $X|x\rangle = x |x\rangle$, where the x is a real number, the spectral value of X.

Soo... $|\psi\rangle = \mbox{(something)} ~ |x\rangle$

What's (something) equal to ?

 

[jinksys]

When you say spectral value, is that the same as the eigenvalue?

 

[dextercioby]

A generalization of eigenvalue. But you can think of it in a simplified way as an eigenvalue.

 

[paranormal]

Great job! Your attempt is correct. In this case, C_n would represent the coefficient of the state vector |psi> in the basis vector |x_n>. This means that to represent |psi> in terms of the continuous basis vectors {|x>}, we would need to find the coefficients C_n for each basis vector. These coefficients could be found by taking the inner product <x_n|psi>. In this way, we can expand the state vector |psi> in terms of the continuous basis vectors, similar to how we would expand it in terms of discrete basis vectors.