Quantum Physics Problems: Waves, Particles, and Energy Levels

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1) A wave has the form

When x = 0, the wavelength is

By applying continuity conditions at x = 0, find the amplitude Ax>0 (in terms of A) and phase Φ of the wave in the region x > 0. Use any variable or symbol stated above as necessary.

2) An electron is trapped in an infinite well of width L = 1.87nm. What are the three longest wavelengths permitted for the electron's de Broglie waves?

The wave function must be zero everywhere the potential is infinite. So the wave function is zero outside the well. Since the wave function must be continuous, the wave function inside the well must go to zero at the edges of the well. Thus, only certain discrete wavelengths are allowed.

3) A particle is described by the wave function ψ(x) = b(a2 - x2) for -a ≤ x ≤ a and

ψ(x) = 0 for x ≤ -a and x ≥ a, where a and b are positive real constants. Using the normalization condition, find b in terms of a. What is the probability to find the particle at x = 0.3a in a small interval width of 0.01a? What is the probability for the particle to be found between x = 0.10a and x = 0.84a?