How Can You Find ψ(x,t) from ψ(x) for a Free Particle?

  • Thread starter JordanGo
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In summary, the conversation discusses finding the coefficient expansion function of the free-particle wavefunction and solving the integral using the error function or other techniques such as completing the square or complex analysis. It is recommended to learn how to solve the integral by hand as the techniques used are useful to know.
  • #1
JordanGo
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Homework Statement


Consider the free-particle wavefunction,
ψ(x)=(pi/a)^(-1/4)*exp(-ax^2/2)

Find ψ(x,t)


The Attempt at a Solution



The wavefunction is already normalized, so the next thing to find is coefficient expansion function (θ(k)), where:

θ(k)=∫dx*ψ(x)*exp(-ikx) from -infinity to infinity

But this equation seems to be impossible to solve without error function (as maple 16 tells me).

Is there any trick to solve this?
 
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  • #2
The error function has very nice properties at the infinity, so you should be able to compute the integral. Alternatively, you could use the apparatus of complex analysis to evaluate the integral.
 
  • #3
Why not use some of the simple Gaussian integral tricks (like completing the square in the exponential)? Or am I missing something?
 
  • #4
You can avoid the error function because of the limits on the integral, or equivalently, use the nice properties of the error function at those limits.

You should really learn to crank this integral out by hand, though. The techniques used are useful to know.
 
  • #5




I would recommend using numerical methods to solve this integral. One possible approach is to discretize the integral and use numerical integration techniques such as Simpson's rule or the trapezoidal rule. Alternatively, you could also try using a computer program specifically designed for solving integrals, such as Wolfram Alpha or Mathematica. Additionally, if the integral cannot be solved analytically, you could try approximating it using a series expansion or using other methods such as Monte Carlo simulation. It is important to keep in mind that sometimes, equations in physics cannot be solved analytically and numerical methods must be used instead.
 

Related to How Can You Find ψ(x,t) from ψ(x) for a Free Particle?

1. How do I find psi(x,t) using psi(x)?

In order to find psi(x,t) using psi(x), you will need to use a mathematical formula called the time-dependent Schrödinger equation. This equation relates the time evolution of a quantum system described by psi(x) to its energy and potential. By solving this equation, you can determine the time-dependent wave function, psi(x,t), which will give you the information you need to find psi(x,t) using psi(x).

2. What is psi(x) and why is it important?

Psi(x) is the wave function that describes the quantum state of a system at a specific point in space. It is important because it contains all the information about the system's energy, momentum, and other physical properties. It also allows us to calculate the probability of finding a particle at a certain position, which is essential in understanding the behavior of quantum systems.

3. Can psi(x) be used to predict the behavior of a quantum system?

While psi(x) contains all the information about a system, it cannot be used to predict its exact behavior. This is because the uncertainty principle states that it is impossible to know both the position and momentum of a particle simultaneously. However, by using psi(x) and other quantum mechanical principles, we can make probabilistic predictions about the behavior of a system.

4. How does psi(x) relate to classical physics?

In classical physics, the state of a system is described by its position and momentum. In quantum mechanics, however, the state of a system is described by the wave function, psi(x). This is because at the quantum level, particles do not have well-defined positions and momenta, but instead behave like waves. Psi(x) captures this wave-like behavior and allows us to make predictions about the behavior of particles in quantum systems.

5. Are there any limitations to using psi(x) to describe quantum systems?

While psi(x) is a powerful tool for understanding quantum systems, it does have some limitations. For example, it cannot be used to describe systems with more than one particle, as it only describes the state of a single particle. Additionally, it cannot be used for systems with strong interactions or in extreme conditions, such as near black holes. In these cases, more advanced mathematical tools and theories are needed to accurately describe the behavior of the system.

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