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# Wavefunction

### Calculations with Wave Functions

1.) A particle of mass m is confined to a one-dimensional potential well with infinite potential walls. The well extends from 0 (less than or equal to) x (less than or equal to) a. At time t = 0, the normalized wavefunction is (see attached file for equation) What is the wavefunction at a later time t = t0? See atta

### Quantum Mechanics Question

The Hamiltonian of a certain three level system is represented by the matrix. See attached file for full problem description.

### Two well problem.

See attached file.

### Ground State of Harmonic Oscillator

In the ground state of the harmonic oscillator, what is the probability (correct to three significant digits) of finding the particle outside the classically allowed region? (Hint: what are the minimum and maximum values of the coordinates of the respective classical oscillator with a given energy E?) Look in math tables under "

### 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

### Relative states and Hidden Variables in Quantum Mechanics

I ONLY NEED HELP WITH NUMBER ONE. It looks really long but the beginning is just a set up to the question. Skip to 'Your job' to see the question. I don't understand this stuff at all so if you could guide me through this step by step it would be appreciated. Explanations are important.

### 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

### 1-D Quantum Mechanics

A basic model of a hydrogen atom is a finite potential well with rectangular edges. A more realistic model of a hydrogen atom, although still a 1-Dimensional model, would be the electron + proton potential energy in one dimension: U(x) = -e^2/(4pi epsilon_0)|x|) a) Draw a graph of U(x) versus x. Center your graph at x = 0

### Wave Functions with Uncertainty

The Wave function of a particle is seen in the attachment. a) Assuming that this function is continuous, what can you conclude about the relationship between b and c? b) Draw graphs of the wave function and the probability density over the interval -2mm <= x <= 2mm. c) What is the probability that the particle will be

### 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

### Mean position in a 1-D harmonic oscillator

Obtain the mean position, <x>, for a particle moving in a 1-D harmonic oscillator potential, when the particle is in the state with normalized wavefunction: Y(x)= ((a/(4*pi))^.25)*(2ax^2-1)*exp((-ax^2)/2)

### 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

### Wave Function: A Linearly Polarized Wave

Please give a step by step solution: 1) The wave function for a linearly polorized wave on a taut string is: y(x,y)=Asin(wt - kx + phi) where A =0.4m, w=3.2s^-1, k=8.1m^-1, phi = 0.49, t is in seconds and x and y are in meters. What is the speed of the wave in m/s? b) What is the vertical displacement of the string

### 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

### Standing waves on string

(See attached file for full problem description and figures) --- Nodes of standing waves: The nodes of a standing wave are points at which the displacement of the wave is zero at all times. Nodes are important for matching boundary conditions, for example, that the point at which a string is tied to a support has zero displ

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?

### Equation of Motion

Find the equation of motion for a particle at x=6.1 for a transverse sine wave with frequency 208 Hz and amplitude .233 meters if the wave propagates in the x directon at 68 m.s, provided the particle at x=0 has equation of motion y=A sin('omega t).

### Harmonic Wave Function - Example

Write an expression for a harmonic wave that has a wavelength of 2.8 m and propagates to the left with a speed of 13.3 m/s. The amplitude of the wave is 0.12 m.

(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

### 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