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# Wavefunction

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

Draw the PIB wavefunctions for 1,3-butadiene.

### 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

### Bound-state wave functions

Why can bound-state wave functions be chosen to be real? The text states that, in one-dimensional problems, the spatial wave function for any allowed state can be chosen to be real-valued. Verify this using the following outline or some other method. Please see the attached document for the questions, listed a) though c).

### One dimensional infinite square, particle with mass

1- Consider a particle with mass m in the one dimensional infinite square ell potential V(x) with length L at time t=0, the wave function for this particle psi (x) =A sin(pi x/L) cos(pi x/L) a) find the coefficient A so that the wave function is properly normalized. b) what is the probability of finding particle between 0<x

### A Spin 1/2 Particle

A spin 1/2 particle is in the state |Psi> = Sqrt[2/3] | Up > + Sqrt[1/3] | Down > Suppose a measurement is made of the spin in the z direction and the result is m_s = -1/2. Now a second measurement is made to determine the spin in the x - direction. What is the probability the spin will be in the +x direction? So I underst

### The momentum wave function for the hydrogen atom

The hydrogen atom ground state may be described by the spatial wave function as: phi(r)= [(1/(Pi*(a_0^(3)))^(1/2)]*e^(-r/a0) This is where a_0 is the Bohr radius. Using the following equation, please calculate the momentum wave function. integral[(e^(-a*r + i*b*r))*(d^(3))*(r) = 8*Pi*a/((a^(2) + b^(2))^(2)) See the at

### Fourier Expansion

Assume a Fourier expansion of the form [see the attachment for equation] and determine the coefficients b_n(t). The initial conditions are [see the attachment for equations]. Note. This is only half the conventional Fourier orthogonality integral interval. However, as long as only the sines are included here, the Strum-Lio

### Expected value of momentum and Fourier transform

The expected value of the momentum operator in quantum mechanics can be calculated using the spatial wave function. Using that the momentum wave function is the Fourier transform of the spatial wave function we obtain an expression for this expected value in term of the momentum wave function, that is, we prove the Equation 15.6

### 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

### Hypothetical wave function for a particle confined in the region

Please see the attached file for the full problem. Consider the following hypothetical wavefunction for a particle confined in the region ... a. What is the value of A? b. Determine the expectation values <x> and <x^2>.

### Two Problems in Modern Physics..

28. A wave packet describes a particle having momentum p. Starting with the relativistic relationship E^2 = p^2 c^2 + E0^2, show that the group velocity is Bc and the phase velocity is c/B (where B = v/c). How can the phase velocity physically be greater than c? 44. An electron microscope is designed to resolve objects as sma

### Problem in Quantum Mechanical Tunneling

Please help me stepwise in word / pdf. Given a tunneling barrier with a thickness of 2nm and a barrier height of 5eV, what is the minimum kinetic energy an electron would have to have to have a 50% chance of passing through? (hint it doesn't necessarily have to be less than the barrier height)

### Eigenvalues and Hamiltonian H

I am confused on how to approach this problem on eigenfunctions and eigenvalues with the Hamiltonian H. Would you please include the work and an explanation, please. Please see the attachment for the full problem. Consider the particle in a box problem for a box of length ... Verify the substitution that the solutions ...

### Eigenfunction Decomposition of 1DHO Wavefunctions

A particle of mass m is subject to the one-dimensional harmonic oscillator potential. Write down the first three normalised eigenfunctions ?_n (x) and the corresponding eigenvalues. Initially the wavefunction is in a mixed state of the form ?(x)=(1/(7???))^(1?2) e^(-x^2/(2?)^2 ) ((3x)^2/(?)^2 +(x/?)-(3/2)+?2) where ?=?(??m?).

### 2D Quantum Mechanical Harmonic Oscillator

A particle of mass m moves in two dimensions under the influence of the potential V(x,y)=1/2 m?^2 (((6x)^2)-2xy+(6y)^2 ). Using the rotated coordinates u=(x+y)/?2 and w=(x-y)/?2 show that the Schrödinger equation in the new coordinates (u,w) is -(?^2)/2m ((d^2/du^2) +(d^2/dw^2))?(u,w)+V ?(u,w)?(u,w)=E?(u,w) Where V ?(u,w) sho

### A particle is trapped in an infinite one-dimensional well of width L. If the particle is in the its ground state, evaluate the probability to find the particle

A particle is trapped in an infinite one-dimensional well of width L. If the particle is in the its ground state, evaluate the probability to find the particle: a) Between x = 0 and x = L/3 b) Between x = L/3 and x = 2L/3 c) Between x = 2L/3 and x = L

### QM Expectation Values for a Particle in a 1D Potential Well

Please see the problem attached I need the step-by-step solution please. Thank you

### Tunneling Probability of an Electron in a Square Well

Given a tunneling barrier with a thickness of 2nm and a barrier height of 5eV, what is the minimum kinetic energy an electron would have to have to have a 50% chance of passing through? (hint it doesn't necessarily have to be less than the barrier height). I need the step-by-step solution please. Thank You.

### Density of States Problem

Identify the quantum state (i.e. list the requisite quantum numbers) of the lowest energy level in a 3-D quantum well that has a zero probability of finding the electron in the center of the structure.

### 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

### Particle Moving in One Dimension

A free particle moving in one dimension is known to be at the point x_1 at time t_1. Assume that the wavefunction at this time is psi(x,t_1) = ?(x - x_1). Find the wavefunction psi(x,t_2) at some later time t_2.

### Calculating the values of Spin Angular Momentum

Calculate the value of <S_x>,<&#8710;S_x>,<S_y>,<&#8710;S_y> ,<S_z> ,<&#12310;&#8710;S&#12311;_z> for the wave function 1/&#8730;2 [exp&#8289;(i&#948;/2) |1/2 1/2>+exp&#8289;((-i&#948;)/2)|1/2-1/2>] Note the labels in the kets are |&#12310;sm&#12311;_s>. (Review question in attachement

### 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: Calculate the probability that at a later time tf the electron spin has flipped to the -z direction.

An electron is in a strong, static, homogenous magnetic field with magnitude B0 in the z direction. At time t=0, the spin of the electron is in the +z direction. At t=0 a weak, homogenous magnetic field with magnitude B1 (where B1 << B0) is turned on. At t=0 this field is pointing in the x direction, but it rotates counterclockw

### 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