General solution for the time-dependent Schrödinger equation

[Dyatlov]

Hello! I have two uncertainties (hehe) about two concepts from a QM time-dependent Schrödinger equation video.
The video is
I cannot move on further if I don't fully grasp everything he explains in the video. My two issues are:
1) The general solution for the time-dependent Schrödinger equation starts at 55:20.He uses αi for a set of coefficients of states. What exactly are these? the probability densities? probability amplitudes? the absolute probabilities? He mentions any state vector can be written as a superposition of eigenvectors of the energy, so α should be a probability amplitude, so you can calculate it's squared modulus.
2) At 1:03:04 he comes up with the exponential solution e^-iEjt. Where exactly from does he gets α(0) from?
Appreciating any kind of answer which would help me shed some light on these two things. Cheers!

 

[blue_leaf77]

1) Usually one calls the square modulus of $\alpha$ as the probability amplitude.
2) Didn't he say that he assumes he knows the wavefunction at t=0, and hence the corresponding expansion coefficients?

 

[Dyatlov]

1) Usually one calls the square modulus of $\alpha$ as the probability amplitude.
2) Didn't he say that he assumes he knows the wavefunction at t=0, and hence the corresponding expansion coefficients?

Thanks for the reply.
1) Fair enough but my question was what does α stand for (he calls it coefficient of states).

 

[blue_leaf77]

If your problem is simply being confused as what to call $\alpha$, you can also adopt what he said in the video. I personally would call it expansion coefficient, the way to obtain it is through the equation $\alpha_i(t) = \langle i | e^{-iHt/\hbar} | \psi(0) \rangle$. I guess you know about it already.

 

[samalkhaiat]

Hello! I have two uncertainties (hehe) about two concepts from a QM time-dependent Schrödinger equation video.
The video is
I cannot move on further if I don't fully grasp everything he explains in the video. My two issues are:
1) The general solution for the time-dependent Schrödinger equation starts at 55:20.He uses αi for a set of coefficients of states. What exactly are these? the probability densities? probability amplitudes? the absolute probabilities? He mentions any state vector can be written as a superposition of eigenvectors of the energy, so α should be a probability amplitude, so you can calculate it's squared modulus.
2) At 1:03:04 he comes up with the exponential solution e^-iEjt. Where exactly from does he gets α(0) from?
Appreciating any kind of answer which would help me shed some light on these two things. Cheers!

When we write |\Psi(t) \rangle = \sum_{n} a_{n}(t) |n\rangle , \ \ \ \ a_{n}(t) = \langle n|\Psi(t)\rangle we are simply representing the abstract state vector |\Psi\rangle by a set of numbers a_{n} which have all the information content in the state Psi. You should know that from vector algebra. Recall the similarity with the vector relations \vec{V}=\sum_{i} v_{i} \hat{e}_{i}, \ \ \ \ v_{i} = \hat{e}_{j} \cdot \vec{V} . So, you say that a_{n}(t) are the components of the “vector” |\Psi\rangle in the |n\rangle “direction”, projection of |\Psi\rangle on |n\rangle, how much of the state |n\rangle one can find in the state |\Psi\rangle, transition (or probability) amplitude from |\Psi\rangle to |n\rangle, or (even better) matrix representation of the state vector |\Psi\rangle. Notice that in a_{n}-representation, the Schrödinger equation becomes i \frac{d}{d t} \langle m | \Psi \rangle = \sum_{n} \langle m |H| n \rangle \ a_{n}(t) , or i \frac{d a_{m}}{d t} = \sum_{n} H_{m n} \ a_{n}(t) . This was the starting equation of the so-called Matrix Mechanics of Heisenberg. It is just an equivalent, matrix form of the differential equation of Schrödinger. Now, suppose that |n\rangle form eigen-states of the Hamiltonian, H|n\rangle = E_{n}|n\rangle, then H_{m n} = \langle m | H | n \rangle = E_{n} \delta_{n m} , and the above matrix equation becomes (no sum on m) i \frac{d a_{m}(t)}{d t} = E_{m} a_{m}(t) . Integrating this from t=0 to t, we find \int_{t=0}^{t} \frac{d a_{m}(t)}{a_{m}(t)} = - i E_{m} \int_{0}^{t} dt ,\ln |\frac{a_{m}(t)}{a_{m}(0)}| = - i E_{m} t , which we normally write as a_{m}(t) = a_{m}(0) e^{ - i E_{m} t}.

 

[Dyatlov]

Thanks a lot, that helped.