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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=

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

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.

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

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

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

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

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?

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?