Quantum Problems: Qs on Eqns 2.49 & 2.50 p15/16

[latentcorpse]

I've attached my notes for this course that I'm taking.
I have a couple of questions about the material at the bottom of p15/top of p16.

Where does eqn 2.49 come from? Why is there an $|x,t \rangle$ on one side and a $| x \rangle$ on the other?

Where does eqn 2.50 come from? It's a rearranging of the line above essentially. But when i left multiply $\hat{H}(t) | x,t \rangle = - i \hbar \frac{\partial}{\partial t} | x,t \rangle$ by $\langle \psi |$, i don't get the right result since i don't think we can move the psi ket through either the hamiltonian operator or the time derivative, can we?

 

[NateD]

Eqn 2.49 comes from the definition of the Schrödinger equation, which states that for any wavefunction ψ(x,t), it must satisfy the equation: \hat{H}(t)|x,t \rangle = -i \hbar \frac{\partial}{\partial t} |x,t \rangle where \hat{H}(t) is the time-dependent Hamiltonian operator. To get Eqn 2.49, we simply rearrange the equation to isolate the wavefunction on one side: |x,t \rangle = e^{-i/\hbar \int^t_{t_0} \hat{H}(t') dt' }|x \rangle Eqn 2.50 is just the result of left multiplying both sides of Eqn 2.49 by \langle \psi | and using the fact that the bra and ket form of a wavefunction are related by a complex conjugate: \langle \psi | x,t \rangle = \langle \psi | e^{-i/\hbar \int^t_{t_0} \hat{H}(t') dt' }|x \rangle = \langle \psi(x) | e^{-i/\hbar \int^t_{t_0} \hat{H}(t') dt' }|x \rangle = \psi^*(x) e^{-i/\hbar \int^t_{t_0} \hat{H}(t') dt' }|x \rangle Therefore, Eqn 2.50 is simply a rearrangement of Eqn 2.49 and is valid since the bra and ket form of a wavefunction are related by a complex conjugate.