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A wave function describes a quantum state of particles and how they behave. The laws of quantum mechanics describe how the wave function changes over time.

The common system for a wave function is Ѱ. Although Ѱ is a complex number, |Ѱ|² is a real number and corresponds to the probability density of finding a particle in a given place at a given time. The SI units for the wave function depend on the system. For one particle in three dimensions the units are m-3/2. The units are required so that an integral of |Ѱ|2 over a region of three-dimensional space is a unitless probability.

The wave function is central to quantum mechanics. It is the fundamental postulate of quantum mechanics. The wave function is a source of the mysterious consequences and philosophical difficulties in the interpretations of quantum mechanics. These topics continue to be in debate today.

Free Particles: Wavefunction, Momentum, and Probability

At t=0, the wavefunction of a free particle is: psi(x,0) = {(sqrt b/2*pi)sin(bx) for |x|<((2*pi)/b) {0 for |x|> or = ((2*pi)/b)) a) What is the probability of finding the particle in the interval 0 less than or equal to x less than or equal to pi/2b? b) What is the momentum am

Fourier transform with complex conjugate

Show that integral from - infinity to + infinity of psi_1(x) times psi_2*(x) dx is equal to integral from - infinity to + infinity of phi_1(k) times phi_2*(k) dk. * indicates complex conjugate

Dimensional Harmonic Oscillator

Consider a 3-dimensional, spherically symmetric, isotropic harmonic oscillator with a potential energy of [see the attachment for full equation]. The Hamiltonian in this case is: [attached] a. Use the trial function [attached] and find the value of the parameter a that the energy, and find that minimum energy. b. Repeat th

Momentum Representation, Momentum Space Wave Function

We study the relationship between the position space wave function and the momentum space wave function in quantum mechanics. We show that they are related by the Fourier transform, more specifically we show that a momentum wave function given by the Fourier transform of the space wave function satisfies the requirements to be a

Problems Involving the Schrodinger Equation

(a) The time-independent Schrodinger equation for a free particle moving in one dimension (x) is written as (d^2 Psi(x) / dx^2) + k^2 Psi(x) = 0, where k is a constant. (i) Show that the following function is a solution to this equation: Psi(x) = A sin(k x) (ii) Briefly outline why constraining the particle within

Nano Wire Problem

Given a Nanowire with cross sectional dimensions of 10 nm x 10 nm, what momentum would an electron in the ground state need in order to possess the same energy as a stationary electron (zero momentum) in the n=1,2 state? Note that, when it says n=1,2 it means nx=1 and ny=2... In reality its not that restrictive, but it wa

Oscillation of pendulum, inertia, amplitude, wave speed

See attached file for proper format. 6. A physical pendulum consists of a uniform rod of mass M and length L. The pendulum is pivoted at a point that is a distance x from the center of the rod, so the period for oscillation of the pendulum depends on x: T(x). (a) What value of x gives the maximum value for T? (Use any va

Triplet/singlet split due to perturbation

Two identical spin-1/2 particles are confined to an infinite one-dimensional square well of width a with infinite potential barriers at x=0 and x=a. The potential is V(x)=0 for 0 <= x <= a. Suppose that the particles interact weakly by the potential V_1(x)=Kdelta(x_1 - x_2), where x_1 and x_2 are the positions of the two particl

Quantum Mechanics: Time dependent perturbation problem

A particle with mass m is in a one-dimensional infinite square-well potential of width a, so V(x)=0 for 0 <= x <= a, and there are infinite potential barriers at x=0 and x=a. Recall that the normalized solutions to the Schrodinger equation are psi_n(x) = sqrt(2/a)sin[(n pi x)/a] with energies E_n = (hbar^2 (pi^2 n^2)/

Time Dependent Perturbation

An electron is in a strong, uniform, constant magnetic field with magnitude B_0 aligned in the +x direction. The electron is initially in the state |+> with x component of spin equal to hbar/2. A weak, uniform, constant magnetic field of B_1 (where B_1 << B_0) in the +z direction is turned on at t=0 and turned off at t=t_0. Let

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

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

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,

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

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

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

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

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

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