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# Wavefunction

### Perturbation Theory

A particle of mass m is in the ground state in the harmonic oscillator potential V(x) = (1/2)Kx^2 A small perturbation (beta)x^6 is added to this potential. a) What are the units of beta? b) How small must beta be in order for perturbation theory to be valid? c) Calculate the first-order change in the energy of the par

### Energy Change of a Particle

A particle of mass m is confined to move in a narrow, straight tube of length a which is sealed at both ends with V=0 inside the tube. Treat the tube as a one-dimensional infinite square well. The tube is placed at an angle theta relative to the surface of the earth. The particle experiences the usual gravitational potential V=

### Probabilities in spin measurements

I have attached the question.

### Physics: Quantum Mechanics

The deuteron is a nucleus of "heavy hydrogen" consisting of one proton and one neutron. As a simple model for this nucleus, consider a single particle of mass m moving in a fixed spherically-symmetric potential V(r), defined by V(r)=-V0 for r<r0 and V(r)=0 for r>r0. This is called a spherical square-well potential. Assume that t

### Particle in a box ground states

The ground state wave function for a particle in a box confined to a 1-D box of length L is &#936;=(2/L)^(1/2) sin (pi x/ L) Suppose that the box is 10.0 nm long. Calculate the probability that the particle is a) between x=9.90 nm and x= 10.0 nm b) in the right half of the box

### Particle in a box function

Particle in a box For the particle in a box, is the function Ψ(x)=Cx(L-x) a possible state of the system? (C is constant) Justify answer. (the parameter l in the wavefunction is the length of the box) V(x)=∞ V(x)=∞

### Particle in one dimensional box

A particle is in the ground state of a box length L. Suddenly the box expands (symmetrically) to twice its size, leaving the wave function undisturbed. Show that the probability of finding the particle in the ground state of the new box is (8/3 pi)^2.

### Time Evolution of Gaussian

Please see the attached file. The problem deals with the series expansion of the time evolution operator and its application to a wave function and show the equivalency between the two expressions.

### Wavefunction normalization, probability

A particle is described by the wavefunction ¦×(x) = A Cos (2¦Ðx/L) for -L/4 ¡Ü x ¡Ü L/4 0 otherwise (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 of its position is made?

### Determining Expectation Value for a Wave Function

The Schrodinger Equation is an important concept in modern day quantum mechanics as there are many systems that involve wave-based particles. Atoms are an example of such a system. Please determine the expectation value of r^2 for a hydrogen atom with the following unnormalized wave function: (see the attached file for the wave

### Energy and Fermi energy of a one-dimensional electrical conductor.

There has recently been considerable interest in one-dimensional electrical conductors. In this problem, you are asked to calculate some free-electron properties for a system of length L containing N electrons. Thus, there are n =N/L electrons per unit length. a) Calculate the density of states per unit energy range per unit

### Normalize Harmonic Oscillator wave functions

The following are two harmonic oscillator wave functions: S(x) = N exp(-max/2h)(2mwx/h) S(x) = N exp(-max/2h)[8(mwx/h)^3 - 12mwx/h] where N = . Show that they are a) normalized, and b) orthogonal

### Quantum Mechanics: Combination of the particle in a box and the Harmonic oscillator.

Please read the attached file for complete description of the problem. Consider the particle of mass subject to a one-dimensional potential of the following form: V(x) = 1/2 kx^2 for x>0 V(x) = + infinity for x < 0 This is a combination of the particle in a box and the harmonic oscillator that might be a better model f

### A particle is in the ground state of a box with sides at x = +/- a.

A particle is in the ground state of a box with sides x = +/- a. Very suddenly the sides of the box are moved to x = +/- b(b > a). What is the probability that the particle will be found in the ground state for the new potential? What is the probability that it will be found in the first excited state? In the latter case,

### Localized particles and energy measurements

A particle is known to be localized in the left half of a box with sides at x = +/- 1/2, with wave function psi (x) = sqrt(2/a) when -a/2 < x < 0 = 0 when 0 < x < a/2 a) Will the particle remain localized at later times? b) Calculate the probability that an energy measurement yields the ground st

### Calculating an Infinite Box Extending Without Schrodinger

Consider an infinte box extending from x=-a/2 to x=a/2. Can you, without actually solving the Schrodinger equation but by looking at the shape of the wave function, immediately give a general formula for the energy eigenvalues for an infinite box that extends for x=-a/2 to x=0? (Hint: Where do the eigenfunctions have nodes?)

### Expectation Value of x^n in a Given Wave Function

Consider the wave function Shi(x) = (alpha/pi)^(1/4) exp(-alpha x^2/2). Calculate <x^n> for n = 1, 2. Can you quickly write down the result for <x^17>? See attachment for further details to the question.

### Show that the expectation value of momentum in the case of a real valued wavefunction is zero.

Please see attached file for full problem statement.

### Wave Function in a Square Well

Problem 1. Consider a square well that extends from 0 to L. (a) Write down the general solution for the wave function inside the well. (b) Determine the specific solutions inside the well for the ground state and for the rst excited state by applying the boundary conditions at x = 0 and at x = L. Now consider a 50:50 superpos

### Speed of Tidal Waves

Show that the speed of tidal waves is given by (gh)^(1/2) where h is the depth of the sea and g is the acceleration due to gravity.

### Measuring Momentum of a Non-Relativistic Particle

Please find the question in the attached document.

### Two-dimensional infinite square well

See attached file.

### Energy & Momentum of a Wavefunction and their Probabilities

The wave function of a particle of mass m that moves in a region of uniform potential V_0 is psi(x) = A(cos kx + cos 2kx), where A and k are constants. Momentum is measured; what are the possible outcomes and their probabilities? Starting with the same initial wave function psi(x) a measurement of the energy is made; what

### Show that the average of the Hamiltonian is an upper bound to the ground state energy.

Show that the average of the Hamiltonian is an upper bound to the ground state energy.

### Two Traveling Waves Beating Together

To see how two traveling waves of nearly the same frequency can create beats and to interpret the superposition as a "walking" wave, consider two similar traveling transverse waves, which might be traveling for example along a string: y1(x,t) = Asin(k1 x - w1 t) and y2(x,t) = Asin(k2 x - w2 t). They are similar because we ass

### Transverse Wave Traveling In A String: Two Wave Velocities

Wave motion is characterized by two velocities: the velocity with which the wave moves in the medium (e.g., air or a string) and the velocity of the medium (the air or the string itself). Consider a transverse wave traveling in a string. The mathematical form of the wave is: y(x,t) = A sin(kx-omega t). 1. Find the slope

### Nodes of a Standing Wave (Sine)

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 displacement at all times (i.e., the point of attachment does not move). Consider a standing wave

### Probability interpretation of the wavefunction.

See attached file.

### Wave Functions and Energy Levels

Consider the particle in the infinite potential well as shown in Figure 2-1 (see attachment). Derive and graph the wave functions corresponding to the four lowest energy levels. Do not normalize wave functions. Find the four lowest energy levels. Use Matlab to graph the wave functions. What happens if the well dimensions increas

### Operations and Commutator Relationships

Please answer question number 8 only. Please see the attached file. Commutators: One of the characteristics of operators if that they generally do not commute. We define the commutator of two operators through the expression [A,B] = [A][B] - [B][A]. Prove the following commutator relationships: (a) [x,p] = i^h, (b) [E,t]