Alternate form of wave equation

In summary, the conversation is about verifying a wave equation for a free particle and rewriting it in a different form. The solution is to rewrite the equation as products of exponentials and pay attention to the signs of the terms. The type of wave is still unknown and needs to be figured out.
  • #1
Daniel Sellers
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Apologies if this question is better posed in the mathematics section, it is for a quantum mechanics class so I decided to post it here:

We are asked to verify that the following equation is a solution to the Schrodinger wave equation for a free particle:

Psi(x,t) = Ae^i(kx-wt) - Ae^-i(kx+wt)

I did that part just fine, but then we are asked to show that this wave equation can be rewritten as

Psi(x,t) = 2AiSinkxe^(-iwt)

and answer the question: What kind of wave of this?

So far the only thing I can come up with is

Psi(x,t) = 2AiSin(kx)cost(wt)

Which doesn't seem compatible at all.

Can anyone show me how this is the same equation? and if you know 'what kind of wave' this is that would be very helpful too.
 
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  • #2
Daniel Sellers said:
Can anyone show me how this is the same equation?

Try rewriting ##e^{i(kx - wt)}## and ##e^{-i(kx + wt)}## as products of exponentials. Pay careful attention to the signs of the ##wt## terms in the exponentials.
 
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  • #3
PeterDonis said:
Try rewriting ##e^{i(kx - wt)}## and ##e^{-i(kx + wt)}## as products of exponentials. Pay careful attention to the signs of the ##wt## terms in the exponentials.

In fact, that works! Thanks very much!

Now I just need to figure out what the problem means when it asks what type of wave this is and I'll be set.
 

1. What is an alternate form of wave equation?

An alternate form of wave equation is a mathematical expression that describes the behavior of a wave in terms of its amplitude, wavelength, and frequency. It is commonly used in physics and engineering to model various types of waves, such as sound waves, electromagnetic waves, and water waves.

2. How does the alternate form of wave equation differ from the traditional form?

The alternate form of wave equation is derived from the traditional form by making certain assumptions and simplifications. It is often written in terms of different variables, such as angular frequency and wave number, and may have a different form depending on the type of wave being modeled.

3. What are the advantages of using the alternate form of wave equation?

One advantage of using the alternate form of wave equation is that it can be applied to a wider range of problems and phenomena. It also allows for easier manipulation and analysis of wave behavior, as it may be more intuitive or simplified compared to the traditional form.

4. Are there any limitations to the alternate form of wave equation?

Like any mathematical model, the alternate form of wave equation has its limitations. It may not accurately describe certain types of waves or may require additional assumptions to be made in order to be applicable. It is important to carefully consider the assumptions and limitations when using this equation.

5. How is the alternate form of wave equation used in real-world applications?

The alternate form of wave equation has a wide range of applications in various fields, such as acoustics, optics, and fluid dynamics. It is used to analyze and design various systems and devices, such as musical instruments, antennas, and water turbines. It is also used in predictive modeling and simulations for practical purposes, such as predicting weather patterns or earthquake behavior.

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