Sunday, February 20, 2011

Mastering Physics: Free Particle Wave Equation

 The Schrödinger equation for a free particle (no potential energy) is
-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi.

Part A
What is the most general solution psi of the time-independent Schrödinger equation?

Part B
What are the energy levels E of this free particle associated with a wave number k?
Express your answer in terms of wave number k, mass m, and Planck's constant divided by 2\pi: hbar.

  E  = \frac{{\hbar}^{2}k^{2}}{2m}

Part C
To normalize this wave function, you must calculate the integral \int_{-\infty}^{\infty}\left|\psi\right|^{2}dx. What is the value of this integral?
\int_{-\infty}^{\infty}\left|\psi\right|^{2}dx =
Thus, a free particle wave function is unnormalizable. This is due to the fact that a free particle wave function has no boundaries and thus is unlocalized. This means that there is the same probability of finding a particle anywhere in the universe. You can think of this in a different way: Since there are no boundaries, there is no potential energy present, and thus a free particle's energy is not constrained. This means that a free wave has a continuous spectrum of frequencies.

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