Solve Schrodinger Equations: Fourier Series/Integral for Psi(x)

In summary, the conversation discusses the wave function for a particle of mass m in one dimension and how to express it in terms of a Fourier series or integral. The wave function is initially given as psi(x)= (const)ex2/4l2, but the participant points out that a negative sign is missing in the exponent. It is then suggested to express the wave function as a Fourier integral with the unknown function f(k). The participant also mentions that they are unsure of the meaning of the variable L in the wave function and that it should resemble a gaussian curve.
  • #1
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Homework Statement



Suppose the wave function for a particle of mass m in one dimension is
psi(x)= (const)ex2/4l2

express this wave function in terms of a Fourier series or integral.

I know that psi(x)=[itex]\sum[/itex]k (ak *eikx) for a plane wave. I have no idea where to go from here. Any help is appreciated. I don't even know what the L in the wave function represents. Psi(x) should look like a gaussian curve.
 
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  • #2
You're missing a negative sign in the exponent. It should be [tex]\psi(x) \propto e^{-\frac{x^2}{4l^2}}[/tex]If you express it as a Fourier integral, you have[tex]\psi(x) = \frac{1}{2\pi}\int_{-\infty}^\infty f(k)e^{ikx}\,dk[/tex]You need to figure out what f(k) is.
 

1. What is the Schrodinger equation?

The Schrodinger equation is a mathematical equation that describes how the quantum state of a physical system changes with time. It is a fundamental equation in quantum mechanics and is used to calculate the probability of finding a particle in a particular state.

2. What is the Fourier series used for in solving the Schrodinger equation?

The Fourier series is used to represent a periodic function as a sum of sine and cosine functions. In solving the Schrodinger equation, the Fourier series is used to represent the wave function (ψ) of a quantum system in terms of its energy states.

3. How is the Fourier integral used in solving the Schrodinger equation?

The Fourier integral is used to represent a non-periodic function as a continuous spectrum of sine and cosine functions. In solving the Schrodinger equation, the Fourier integral is used to represent the wave function (ψ) for a non-periodic potential.

4. What is the significance of the wave function (ψ) in the Schrodinger equation?

The wave function (ψ) in the Schrodinger equation represents the probability amplitude of finding a particle in a particular state. It describes the behavior of a quantum system and is used to calculate the probability of finding a particle in a specific location or energy state.

5. How does solving the Schrodinger equation using Fourier series/integral help in understanding quantum mechanics?

Solving the Schrodinger equation using Fourier series/integral allows us to understand the behavior of quantum systems and make predictions about their properties. It provides a mathematical framework for studying the wave-like nature of particles and the probabilistic nature of quantum mechanics.

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