When can the time-independent Schrodinger be used?

In summary, the conversation discusses the use of the Schrodinger equation in periodic solids and the assumption that the time factor can be neglected due to no energy exchange with the environment. The potential landscape and potential energy are also mentioned, along with the use of the time-independent version of the equation for analyzing the spatial part of the wavefunction. Additionally, the concept of coherent electron movement through the lattice is brought up.
  • #1
sokrates
483
2
Hi,

I am new to this forum. I realized that I was browsing the forums way too much and I said why not register and post some questions that have been lingering in my head:

Here is one:

In a periodic solid, we almost always neglect the time factor of the Schrodinger equation:

[tex]e^{-i Et/hbar}[/tex]

So I guess the implicit assumption here is that there's no energy exhcange with the environment (no phonons, etc...) so that the total energy of the electron remains the same?
But what about the potential landscape the electron sees? The potential due to lattice atoms is changing periodically.

So the potential energy [tex]U(x) [/tex] in the Schrodinger equation is changing.

How can we understand this then?
 
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  • #2
If the potential U(x) is a function of x alone (and not time), then Schrodinger's equation separates, and we may use the the time-independent version to analyze the spatial part of the wavefunction.
 
  • #3
I guess we should add that "if the electron is moving [tex]\textbf{coherently}[/tex] through the lattice..."

But that something we almost always assume, right?

Thank you for the response.
 

1. What is the time-independent Schrodinger equation?

The time-independent Schrodinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system in terms of its energy and potential. It is used to determine the allowed energy states and corresponding wave functions of a system.

2. When can the time-independent Schrodinger equation be used?

The time-independent Schrodinger equation can be used when the potential energy of a system does not change with time. This is known as a stationary system, and it allows for a simplification of the equation and easier solutions to be found.

3. What is the difference between the time-independent and time-dependent Schrodinger equations?

The time-independent Schrodinger equation is used for stationary systems, while the time-dependent Schrodinger equation is used for systems where the potential energy is changing with time. The time-dependent equation takes into account the time evolution of the system's wave function.

4. Can the time-independent Schrodinger equation be used for all quantum systems?

No, the time-independent Schrodinger equation can only be used for quantum systems where the potential energy is not changing with time. For systems with time-varying potentials, the time-dependent Schrodinger equation must be used.

5. How does the time-independent Schrodinger equation relate to the concept of quantization?

The time-independent Schrodinger equation is derived from the principle of quantization, which states that energy and other physical properties can only exist in discrete, quantized values. The equation helps to determine these quantized energy states and their corresponding wave functions for a given system.

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