Sketch a diagram to show a comparison of energy levels and wavefunctions for a quantum particle between a rigid box and a finite square well at the n=1 and n=2.
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Sketch a diagram to show a comparison of energy levels and wavefunctions for a quantum particle between a rigid box and a finite square well at the n=1 and n=2.

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- Solve the time-independent Schrödinger equation and determine the energy levels and the wave function of a particle in the potential a? V (x) = Vol a + 2r2 with a = const.For quantum harmonic insulators Using A|0) = 0, where A is the operator of the descending ladder, look for 1. Wave function in domain x: V(x) = (x|0) 2. Wave function in the momentum domain: $(p) = (p|0)Consider a particle moving in a one-dimensional box with walls at x = -L/2 and L/2. (a) Write the wavefunction and probability density for the state n=1. (b) If the particle has a potential barrier at x =0 to x = L/4 (where L = 10 angstroms) with a height of 10.0 eV, what would be the transmission probability of the electrons at the n = 1 state? (c) Compare the energy of the particle at the n= 1 state to the energy of the oscillator at its first excited state.
- Consider a particle in the n = 1 state in a one-dimensional box of length a and infinite potential at the walls where the normalized wave function is given by 2 nTX a y(x) = sin (a) Calculate the probability for finding the particle between 2 and a. (Hint: It might help if you draw a picture of the box and sketch the probability density.)An electron is trapped inside a 1.00 nm potential well. Find the wavelength of the photons when the electron makes a transition from n =4 to n= 1.A particle is trappend in a one-dimensional well. Two of its wavefunctions are shown below. (a) Identify wether the well is finite or infinite. (b) Identify the quantum number n associated with each wavefunction; (c) Overlay a sketch of the probability density for each wavefunction. n = n =