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# Wavefunction

### Normalization of a wavefunction.

See question 6 of the attached file.

### Sinusoidal wave propagating along a stretched string

A sinusoidal wave is propagating along a stretched string that lies along the x-axis. The displacement of the string as a function of time is graphed in the figure for particles at x = 0 and at x = 0.0900 m. What is the amplitude of the wave? What is the period of the wave? You are told that the two points x = 0 and x = 0.0

### Drawing a Graph for Given Curves

A)Draw a graph to represent the currents I = 2.3 sin wt and I = 2.9 sin (wt + 35degrees) B)ON the same graph draw the form of the current when I = 2.3 sin wt and I = 2.9 sin (wt + 35degrees) are added together. C)Express the resultant wave. D)From the graph, what is the amplitude of the resultant wave when the angle is at; I

### Guitar String Tension Effects on Frequency/Oscillation

This is a conceptual question. The tension in a guitar string is doubled. Does the frequency of oscillation also double? If not, by what factor does the frequency change? Specify whether the change is an increase or a decrease.

### Constructive and Destructive Interference

First pulse has wavelength 50 and amplitude 1.0. Second pulse has wavelength to 50 and the amplitude to 1.0. Move the wave and describe why they behave as they do.

### Quantum Mechanics and Particle Position

Part A) What is the expectation value of finding the particle at x = 2L/3 in a box of length L and in the ground state? Part B) Same question with the particle at the first excited state.(n=2) Part C) Can you explain what is the difference between a probability and an expectation value? (4 of the 7 are for the explanati

### Quantum Mechanics

A particle is in the first excited state (n = 2) of a box of length L. Find the probability of finding the particle in the interval deltax = 0.002 located at x = L/2 .

### Probability Calculation of Ground State

A particle is in the ground state of a box of length L. Find the PROBABILITY of &#64257;nding the particle at x = 2L/3. This is a number between 0 and 1

### Infinite square well and Schroedinger equation

Show a detailed derivation for the time independent Schroedinger Equation for an 1-d square well , symmetric about the origin, with a dimension of L (-L/2 to 0 to +L/2).

### Wavefunction Normalization

Often the relative probability of finding an atom in its excited state at time t is given by |psi(t)|^2 ~ e^(-2t/T), where T is the lifetime of the excited state. Normalize this probability distribution, and when does the probability drop to half the maximum value?

### Attractive Delta Function Potential with Step Function

See the attached filled.

### Radius of the hydrogen atom

The "radius of the hydrogen atom" is often taken to be on the order of about 10^-10m. If a measurement is made to determine the location of the electron for hydrogen in its ground state, what is the probability of finding the electron within 10^(-10) m of the nucleus?

### Time Dependent Wavefunctions and Derivatives

(a) Let Q be an operator which is not a function of time, and let H be the Hamiltonian operator. Provide proof for an equation (see attached file for equation). Here {q} is the expectation value of Q for an arbitrary time-dependent wae function , which is not necessarily an eigenfunction of H, and {[Q,H]} is the expectatio

### Three-Dimensional Time-Independent Schrodinger

A particle with mass m is confined inside of a spherical cavity of radius ro. The potential is spherically symmetric and can be written in the form: V(r)=0 for r<ro, and V(r)=infinity for r=ro. The particle is in the l=0 state. (a) Solve the radial Schrodinger equation and use the appropriate boundary conditions to find the g

### Hydrogen Atom and Electrons

a) The electron in a hydrogen atom is in the l=1 state having the lowest possible energy and the highest possible value for m1. What are the n, l, and m1 quantum numbers? b) A particle is moving in an unknown central potential. The wave function of the particle is spherically symmetric. What are the values of l and m1?

### 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.

### One-Dimensional Time-Independent Schrodinger

4.15. A particle is moving in a simple harmonic oscillator potential V(x) = 1/2kx^2 for x>0, but with an infinite potential barrier at x=0 (the paddle ball potential). Calculate the allowed wave functions and corresponding energies. 4.16. A particle moves in one dimension in the potential V(x) Vo ln(x/xo) for x>0, where xo an

### 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

### Momentum space representation of a wavefunction.

See attached file.

### 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

### Mixed State in Infinite Well

Mixed state in infinite well. See full description in the attached file.

### Hydrogen Atom Ground State, and Uncertainty Relation

Hydrogen atom ground state, uncertainty relation See full description in the attached file.

### 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))

### Commutator of Wave Function

Find the commutator of [x , d/dx ] for the wave function described in the attachment. See attached file for full problem description.