Time-Dependent Schrodinger Eq: Integrating Time?

[Pythagorean]

My own question, carried to a new thread as not to derail another:

This brings another question to mind, regarding whether that information (time) is already implicitly present [in the time-independent Schroedinger equation]. In modern physics is it as easy to integrate time into an equation as it is with classical physics?

My line of thought being that if you have the Hamiltonian, you have momentum, which contains velocity, which can be separated and integrated as dx/dt to introduce a time-dependence.

I've seen the formula for the time-dependent equation, but I haven't followed the mathematical development of it. I've only really worked with the one-dimensional, time-independent schrodinger equation... and the math is still shaky for me.

 

[genneth]

Volumes have been written on this subject -- google for it, and focus on the papers from peer reviewed journals -- there's quite a bit of crack all over on this topic. You might want to learn classical mechanics a little further -- Goldstein, Classical Mechanics is the classic (no pun intended) reference. Understanding the role of time in classical mechanics is a prerequisite before you move on to the much trickier situation in QM. Suffice to say that momentum and velocity have almost exactly nothing to do with each other.

 

[denian]

The short answer is yes, it is possible to integrate time into equations in modern physics. However, the process may not be as straightforward as in classical physics.

In classical physics, time is often treated as an independent variable and can easily be integrated into equations. However, in quantum mechanics, time is treated as a parameter and is not necessarily an independent variable. This means that it cannot simply be integrated in the same way as in classical physics.

In order to incorporate time into the Schrodinger equation, one must use the time-dependent Schrodinger equation, which takes into account the time evolution of a quantum system. This equation is derived from the time-independent Schrodinger equation, but includes a time-dependent term known as the time evolution operator.

Integrating time into quantum mechanics also requires a more sophisticated understanding of mathematical concepts such as complex numbers and operators. This can make it more challenging for some to follow the mathematical development of the time-dependent Schrodinger equation.

Overall, while it is possible to integrate time into equations in modern physics, it may not be as straightforward as in classical physics and may require a deeper understanding of mathematical concepts.