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Question about Nodes of a Standing Wave

(See attached file for full problem description) --- Nodes of a Standing Wave (Cosine) Learning Goal: To understand the concept of nodes of a standing wave. The nodes of a standing wave are points where the displacement of the wave is zero at all times. Nodes are important for matching boundary conditions, for example that

Values of the radius

Find the lesser and greater values of the radius where the n = 2, l = 0 radial probability density has its maximum values.

Radial Wave Function

(a) Use this reursion formula, c_j+1 = (2(j+l+1-n)*c_j)/((j+1)(j+2l+2)), to confirm that when l=n-1 the radial wave function takes the form: R_n,n-1 = (N_n)*r^(n-1)*e^(-r/(na)) (b) Calculate <r> and <r^2> for states psi_n,n-1,m.

Probability of Finding Particles

If a hydrogen atom is in the ground state, what is the probaility of finding the electron in a volume of 1.0 pm^3 at a distance of 52.9 pm from the nucleus, in a fixed but arbitrary direction? 1 pm = 10^-12 m

Electron Tunneling

I am given three unnormalized wavefunctions for the system: psi(x) = 100e^x for x<-4 psi(x) = 0.73 cos[(pi)x/40] for-4<x<4 psi(x) = 100e^-x for x>4 I need to determine the probability of the wavefunction vs. x for this system from x=-10 to x=10 so that I can plot it. I have to comment on the probability of fin

Energy quantization

Derive the infinite square well energy quantization law, directly from the DeBroglie relation p=h/l, by fitting an integral number of half DeBroglie wavelengths l/2 into the width a of the well

Entangled states of wavefunctions.

Suppose that a pair of electrons, A and B, were described by the following wave function: (see attached for equations). (I have rewritten this equation as I believe some of you are having problems reading the text.) What property specific to entanglement must the wavefunction describing an entangled state of two particles

Schrodinger Equation for a Harmonic Oscillator

The schroedinger equation for harmonic oscillator can be written: E*psi = [(h^2)/2m][((d^2)*psi)/(dx^2)] + (1/2)kx^(2*psi) Write and formally differentiate each term to get the second derivative with respect to X. Put it all into the equation as shown and you will see that there will be an infinite number of possible solu

Fermions in harmonic potential.

Two identical, non-interacting spin-1/2 fermions are placed in the 1-D harmonic potential V(x) = (1/2)m ω2x2, Where m is the mass of the fermion and ω is its angular frequency. a. Find the energies of the ground and first excited states of this two-fermion system. Express the eigenstates corresponding to these two