Finding the wavefunction

In summary, in this conversation, the problem is to construct the function \psi (x,t) given the function a(k) = \frac{C \alpha}{\sqrt{ \pi}} e^{-\alpha ^2 k^2} and constants alpha and C. The process involves pulling out the constants from the given function and joining them with exponentials. The next step is to evaluate the integral, which can be tricky. Some possible approaches include completing the square and expressing the complex exponential in terms of sine and cosine. It may also be helpful to use an online calculus tool to look up the integral. The conversation also mentions the possibility of moving this topic to the advanced physics section and inquires about the LaTeX
  • #1
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In this problem I am given a function [tex] a(k) = \frac{C \alpha}{\sqrt{ \pi}} e^{-\alpha ^2 k^2} [/tex]
where alpha and C are both constants

Now I am supposed to construct [tex] \psi (x,t) [/tex]

My work:
[tex]\psi (x,t) = \int_{-\inf}^{\inf} a(k) e^{i[kx - \omega (k) t]} dk [/tex]
pull out the constants from our given function and join exponentials to get
[tex]\psi (x,t) = \frac{C \alpha}{\sqrt{\pi}} \int_{-inf}^{inf} e^{i[kx - \omega (k) t] - \alpha ^2 k^2} dk [/tex]
Here's where I am unsure. This is strange integral to evaluate, but my tactic was to complete the square, and hope for the best. The rest of my work is shown below, but I don't know if it is right or not
[tex]\psi (x,t) = \frac{C \alpha}{\sqrt{\pi}} \int_{-inf}^{inf} e^{-(\alpha k - kx/2)^2 - (kx/2)^2 - \omega (k)t} dk [/tex]
Where do I go from here? What is the best way to evaluate this integral?
 
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  • #2
Maybe quantum mechanics isn't intro level physics, should this be moved to advanced physics?

Also, what is the LaTeX command for infinity?
 
  • #3
Mindscrape said:
Maybe quantum mechanics isn't intro level physics, should this be moved to advanced physics?

Also, what is the LaTeX command for infinity?
It probably should be in the advanced section, but it is here now.

Have you tried expresseing the complex exponential in terms of sine and cosine? Have you tried using an online calculus tool to look up the integral?

[tex] \infty [/tex]

Do you have a finctional form for [tex] \omega (k) [/tex]?
 
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1. What is the wavefunction in quantum mechanics?

The wavefunction, also known as the quantum state, is a mathematical description of the quantum state of a particle or system. It contains all the information about the position, momentum, and other physical properties of the particle or system.

2. How is the wavefunction used in quantum mechanics?

The wavefunction is used to calculate the probability of obtaining a certain measurement result in quantum mechanics. By applying mathematical operations to the wavefunction, such as the Schrödinger equation, scientists can predict the behavior and properties of particles and systems.

3. How is the wavefunction related to the uncertainty principle?

The uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a particle. This is due to the wave-like nature of particles described by the wavefunction. The more accurately we know the position of a particle, the less accurately we can know its momentum, and vice versa.

4. Can the wavefunction be observed or measured?

No, the wavefunction itself cannot be observed or measured. It is a mathematical concept used to describe the quantum state of a system. However, the effects of the wavefunction can be observed through measurement and experimentation.

5. How does the wavefunction collapse in quantum measurements?

When a measurement is made on a quantum system, the wavefunction "collapses" to a particular state. This means that the probability of obtaining a certain measurement result is reduced to 100%. The exact mechanism of this collapse is still a topic of debate among scientists.

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