Understanding QM Proof: Wavefunction in Orthonormal Eigenfunctions

[Silviu]

Hello! I have a proof in my QM book that: $\left<r|e^{-iHt}|r'\right> = \sum_j e^{-iHt} u_j(r)u_j^*(r')$, where, for a wavefunction $\psi(r,t)$, $u_j$ 's are the orthonormal eigenfunctions of the Hamiltonian and $|r>$ is the coordinate space representation of $\psi$. I am not sure I understand this. Like in general $\psi = \sum_j a_j u_j$ and I don't know where the $a_j$'s disappeared. Also I am not sure what is the relation between $u_j(r)$ and $u_j(r')$. Is there any formula linking them? Thank you!

 

[kith]

Have a look at the following and check if you understand it. Note what happens with the exponent - you got this wrong in your post.
$\begin{eqnarray*} \langle r|e^{-iHt}|r'\rangle &=& \langle r|e^{-iHt} \left( \sum_j |E_j\rangle \langle E_j| \right) |r'\rangle\\ &=& \langle r| \sum_j e^{-iHt} |E_j\rangle \langle E_j |r'\rangle\\ &=& \langle r| \sum_j e^{-iE_jt} |E_j\rangle \langle E_j |r'\rangle\\ &=& \sum_j e^{-iE_jt} \langle r|E_j\rangle \langle E_j|r' \rangle\\ &=& \sum_j e^{-iE_jt} u_j(r) u_j^{*}(r') \end{eqnarray*}$

 

[Silviu]

Have a look at the following and check if you understand it. Note what happens with the exponent - you got this wrong in your post.
$\begin{eqnarray*} \langle r|e^{-iHt}|r'\rangle &=& \langle r|e^{-iHt} \left( \sum_j |E_j\rangle \langle E_j| \right) |r'\rangle\\ &=& \langle r| \sum_j e^{-iHt} |E_j\rangle \langle E_j |r'\rangle\\ &=& \langle r| \sum_j e^{-iE_jt} |E_j\rangle \langle E_j |r'\rangle\\ &=& \sum_j e^{-iE_jt} \langle r|E_j\rangle \langle E_j|r' \rangle\\ &=& \sum_j e^{-iE_jt} u_j(r) u_j^{*}(r') \end{eqnarray*}$

Thank you for your reply. I understand the logic of it. However I am a bit confused. From what I see $\|E_j>$ is an eigenfunction of the hamiltonian, but $u_j$ is that, too. What is the difference between them?

 

[kith]

$|E_j \rangle$ is an eigenstate of the Hamiltonian, i.e. it is a vector in Hilbert space.

$u_j(r) = \langle r | E_j \rangle$ is a representation of the state $|E_j \rangle$ which uses the position basis. You can use many different bases to represent the same state vector. $\tilde u_j(p) = \langle p | E_j \rangle$ for example uses the momentum basis and refers to the same state $|E_j \rangle$.