# General solution for the time-dependent Schrödinger equation

• Dyatlov
In summary, the conversation is about two uncertainties regarding the concepts of time-dependent Schrödinger equation and the use of α as coefficients of states. The first issue is about what exactly α stands for, with the suggestion that it could represent probability amplitudes or squared modulus. The second issue is about the exponential solution e^-iEjt and where the α(0) term comes from. It is explained that α represents the abstract state vector and is used to represent the state in the |n> direction. It is also mentioned that in this representation, the Schrodinger equation takes the form of the Matrix Mechanics of Heisenberg. The discussion concludes with the explanation that α(0) comes from integrating the matrix equation from
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!

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?

Last edited:
blue_leaf77 said:
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?
1) Fair enough but my question was what does α stand for (he calls it coefficient of states).

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.

Dyatlov said:
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 Schrodinger 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 Schrodinger. 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 $$a_{m}(t) = a_{m}(0) e^{ - i E_{m} t}.$$

ShayanJ and vanhees71
Thanks a lot, that helped.

## What is the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. It was first derived by Erwin Schrödinger in 1926 and is written as a partial differential equation.

## What is the general solution for the time-dependent Schrödinger equation?

The general solution for the time-dependent Schrödinger equation is given by the wave function, which is a complex-valued function that describes the quantum state of a system. It is a solution to the Schrödinger equation and can be used to calculate the probability of finding a particle in a certain position or momentum.

## What factors affect the time evolution of a quantum system?

The time evolution of a quantum system is affected by various factors, including the initial state of the system, the potential energy of the system, and the external forces acting on the system. These factors can cause the wave function to change over time, leading to different outcomes for the system.

## How is the general solution for the Schrödinger equation related to the uncertainty principle?

The general solution for the Schrödinger equation is related to the uncertainty principle through the Heisenberg uncertainty principle, which states that it is impossible to know both the position and momentum of a particle with absolute certainty. The wave function, which is the general solution, contains information about both the position and momentum of a particle, but it is subject to uncertainty and can only provide probabilities of these quantities.

## Can the general solution for the Schrödinger equation be used to predict the exact behavior of a quantum system?

No, the general solution for the Schrödinger equation cannot be used to predict the exact behavior of a quantum system. The uncertainty principle and the probabilistic nature of quantum mechanics mean that the exact behavior of a system cannot be determined. The general solution can only provide probabilities for the outcomes of measurements on the system.

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