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# Wavefunction

### Particle in an infinite tube

a) A particle with mass m and energy E is inside a square with tube infinite potential barriers at x=0, x=a, y=0 and y=a. The tube is infinitely long in the z direction. Inside the tube v=0. The particle is moving in the +z direction solve the Schrodinger equation to derive the allowed wave functions for this particle. Do not tr

### Semi-infinite square well.

Consider the semi-infinite square well given by V(x)=-Vo<0 for 0<=x<=a and V(x)=0 for x>a. There is an infinite barrier at x=0. A particle with mass m is in a bound state in this potential energy E<=0. a) solve the Schroedinger Eq to derive phi(x) for x=>0. Use the appropriate boundary conditions and normalized the wave funct

### Finding the ground-state radial wave function R(r) and the ground-state energy.

A Particle of mass m is in a three-dimensional spherically-symmetric harmonic oscillator potential given by V(r)=(1/2)Kr^2 The particle is in the l=0 state. Find the ground-state radial wave function R(r) and the ground-state energy.

### Zero Point Energy

A baseball is confined between two thick walls at a distance. 0.5M a part. calculate the zero point energy of the baseball. My book has no examples for zero point energy and it might be on my test. I need a outline of the solution please.

### 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))

### Two well problem.

See attached file.

### Harmonic oscillator problems.

See attached file.

### 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

### Estimate the ground state energy of a particle in the potential V(x) = lambda *(x)^4 using variational methods and the uncertainty principle

Estimate the ground state energy of a particle of mass m moving in the potential V(x) = lambda *(x)^4 by two different methods. a. Using the Heisenberg Uncertainty Principle; b. Using the trial function psi(x)=N*e^{[- abs(x)]/(2a)} where a is determined by minimizing (E) *Note abs = absolute value

### 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

### Semi-infinite potential: Derivation of the transcendental equation for the case of E < Vo and finding the reflection coefficient for the case of E > Vo

A semi-infinite potential well is given as shown in the figure. ---------- Figure ------------------- (a) Consider the case when (0<E<Vo).Show the quantization of energy is given by the following transcendental equation: --------Equation -------------------- (b) A particle of energy E> Vo is incident from the rig

### 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

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

### 'Quantum Mechanics'

This question is from the text book 'Quantum Mechanics' second edition by David J. Griffiths. (See attached file for full problem description) --- Problem 3.5 - The hermitan conjugate (or adjoint) of an operator Qprime... Problem 3.7 - The Hamiltonian and the initial state for a certain three-level system is given. Find

### 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?

(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

1. A 100-keV x ray is Compton-scattered through an angle of 90 degrees. What is the energy of the x ray after scattering? a. 83.6 keV b. 121 keV c. 114.5 keV d. 100 keV 2. What is the de Broglie wavelength of a particle moving at a speed of 1.00 x 10^6 m/s if it is (a) an electron (b) a proton? (me= 9.11

### 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

### {21.24} A 3kg particle has a velocity of (3i-4j) m/s. Find the magnitude of its momentum?

A 3kg particle has a velocity of (3i-4j) m/s. Find the magnitude of its momentum? ============================================= Answers: a) 9kg m/s b) -12 kg m/s c) 15 kg m/s d) 3 kg m/s ################################################## A sinusoidal wave is described by y=(0.30m)sin(0.20x-40t). Determine the wave speed.

### 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