Quantum Physics Problems: Waves, Particles, and Energy Levels

1) A wave has the form

When x = 0, the wavelength is

By applying continuity conditions at x = 0, find the amplitude A_x>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?

4) In a certain region of space, a particle is described by the wave function ψ = Cxe^-bx where C is a real constant, b = 0.9, and m = 2.4. By substituting into the Schrodinger equation, find the potential energy (not necessarily constant) in this region and also find the energy of the particle. (Hint: your solution must give an energy that is constant everywhere in this region, independent of x.)

5) A particle is represented by the following wave function. A) Use normalization condition to find C? B) Evaluate the probability to find the particle in an interval of width 0.01 at x = 0.4 (that is between x = 0.395 and x = 0.405 NO INTEGRATION NEEDED) C) Evaluate the probability to find the particle between x = 0.18 and x = 0.28. D) Find the average values of x and x^2, and the uncertainty of x: Δx = sqrt(x^2bar - xbar^2 )

6) A particle of an infinite well is in the ground state with an energy of 1.64eV. How much energy must be added to the particle to reach the fourth excited state (n = 5)? The eighth excited state (n = 9)?

7) An electron is trapped in an infinitely deep 1D well of width L = 0.297nm. Initially, the electron occupies the n = 4 state. Suppose the electron jumps to the ground state with the accompanying emission of a photon. What is the energy of the photon?

8) A particle is trapped in an infinite 1D well of width L. If the particle is in its ground state, evaluate the probability to find the particle between x = 0 and x = L/3

9) What is the next level (above E = 50E₀) of the 2D particle in a box in which the degeneracy is greater than 2?