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Wavefunction

Semi-Infinite Square Well Potential

Consider the semi-infinte square well potential shown in diagram of attachment. a. Given that the energy eigenvalue for the first excited state E2=0 (i.ie u2 is at the edge of being bound) find the width a of the well in terms of m, h and Vo. (Hint consider the continuity of u2 at x=a). b. Calculate the energy eigenvalue E1 fo

Orbital Radius of the Proton's Wavefunction

In a hydrogen atom, the electron and proton can be considered to 'orbit' each other about their common center of mass. a. If the electron wavefunction has a mean orbital radius equal to 0.527 Angstroms (the Bohr radius) what is the mean orbital radius of the proton's wavefunction? b. If the hydrogen atom is suddenly ionized (i

Particle of mass m in a one-dimensional impenetrable box

A particle of mass m is in a one-dimensional impenetrable box extending from x = 0 to x = a. At t = 0, its wavefunction is given as: (SEE ATTACHMENT FOR EQUATION) a) What is the probability for finding the particle in the ground state at t = 0? b) What is the probability of finding the particle between x = 0 to x = al2 a

Transverse Waves Traveling on a String and Standing Waves

Two transverse waves traveling on a string combine to a standing wave. The displacements for the traveling waves are Y1(x,t) =0.0160 sin (1.30m^-1 x - 2.50 s^-1t +.30). Y2(x,t) = .0160 sin (1.30m^-1 x +2.50s^-1 + .70), respectively, where x is position along the string and t is time. A. Find the location of the first antinod

Standing wave as sum of wave traveling to left and right

See attached file for full problem description. 9.2 Show that the standing wave f(z,t) = Asin(kz)cos(kvt) satisfies the wave equation, and express it as the sum of a wave travelling to the left and a wave traveling to the right as shown in the following equation: F(z,t) = g(z -vt) + h(z + vt)

show that functions satisfty the wave equation

See attached file for full problem description. By explict differentiation, check that the functions f1, f2 below satisfy the wave equation. Shwo that f4 does not. f1(z, t) = Ae^(-b(z-vt)^2) f2(z, t) = Asin[b(z -vt)] f4(z, t) = Ae^(-b(bz^2 + vt))

Energy StD in a Superposition of Two Stationary States

Calculate the standard deviation of the energy for a particle in a state, which is a superposition of two stationary states with coefficients c1 and c2. Do this calculation in two ways: (i) using the wave function of this state and a standard deviation of quantum mechanical averages, and (ii) using the probabilistic interpret

A particle moving in a delta potential with positive energy

A particle of mass m, with energy E>0, is moving in the potential V(x)=g[delta(x+a) + delta(x-a)] Assuming that the particle is incident from the left, what is the solution of the Schrodinger equation in all three regions (x<a, -a<x<a, x>a) for this situation? Also, what are the appropriate continuity conditions at x=+a and

Transverse Wave Properties

A uniform rope of mass m and length L hangs from a ceiling. (a) Show that the speed of a transverse wave in the rope is a function of y, the distance from the lower end, and is given by v = Sqrt(gy). (b) Show that the time it takes a transverse wave to travel the length of the rope is given by t = 2sqrt(L/g). (Hint: calculat

Commutation relations and the uncertainty principle: normalized wave function

View the attached file for proper formatting of formulas. Consider two hermitian operators A and B which satisfy the following commutation relation: [A, B] = AB-BA=iC, where C is also a hermitian operator in general. Let us introduce a new operator Q defined by: Q=A+ i&#955;B, with &#955; being a real number, and consider

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.

Free electron wave function

A free electron has a wave function given by &#968; = A sin(1.0 x1010 x). x is given in meters. What are its a) wavelength b) momentum c) speed d) kinetic energy e) what would the wave function of a free proton traveling at the same speed be?

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.

Matter wave problem - Normalization constant

Please see the attached file for full problem description. 6. A particle is described by the wavelength function: (see attachment) (a) Determine the normalization constant A. (b) What is the probability that the particle will be found between x = 0 and x = L/8 if a measurement is made?

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

Ehrenfest's theorem

Consider a 1-D free particle, describable as a wave packet at initial time t0. a) Show, applying Ehrenfest's theorem, that <X> is a linear function of time and <P> is a constant. b) Write the equations of motion for the mean values <X^2> and <XP + PX>. Integrate these equations. c) Show that, with a suitable choice of t

Normalization of a wavefunction: Example problem

A quantum system has a measurable property represented by the observable S with possible eigenvalues nħ, where n = -2, -1, 0, 1, 2. the corresponding eigenstates have normalized wavefunctions Ψn. the system is prepared in the normalized superposition state given by, *Please see attached for equation* Where N is a normalizin

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

The Intercepts for Time and Location on a Sine Curve

See attached file. A traveling wave on a wire is expressed by the equation: (1) y= .24 sin (11x - 16t). Distances are in meters, times in seconds. PART a. On a general sine curve that you see in ATTACHMENT #1, Show a properly located y axis for the graph of y(x) at t= .25 sec. Calculate and label the y intercept and thr