Consider a particle of mass m that can move in a one-dimensional box of size L...

Consider a particle of mass m that can move in a one-dimensional box of size L with the edges of the box at x=0 and x = L. The potential is zero inside the box and infinite outside.

You may need the following integrals:

∫ 0 1 d y sin ⁡ ( n π y ) 2 = 1 / 2 , for all integer n

∫ 0 1 d y sin ⁡ ( n π y ) 2 y = 1 / 4 , for all integer n

∫ 0 1 d y sin ⁡ ( n π y ) 2 y 2 = 1 6 − 1 16 π 2 , for all integer n

(a) Write down the time-independent Schrodinger equation describing the system, and determine whether ψ = sin ⁡ ( 2 π x / L )is an energy eigenstate by checking whether it satisfies the Schrodinger equation. If it is an energy eigenstate, what is the energy? [10 pts]

(b) Suppose we measure the position of the particle in the state in part (a) repeatedly and average the results. Determine the result. [10 pts]

(c) What are the most likely places to find the particle in a single measurement if it is in the state in part (a)? [10 pts]

Expert Solution

genius_generous answered 2 years ago

Consider a particle of mass m confined to a one-dimensional box of length L and in...

Consider a particle of mass m confined to a one-dimensional box of length L and in a state with normalized wavefunction. For a partide in a box the energy is given by En = n2h2/8mL2 and, because the potential energy is zero, all of this energy is kinetic. Use this observation and, without evaluating any integrals, explain why < px2>= n2h2/4L2

Consider the particle-in-a-box problem in 1D. A particle with mass m is confined to move freely...

Consider the particle-in-a-box problem in 1D. A particle with mass m is confined to move freely between two hard walls situated at x = 0 and x = L. The potential energy function is given as (a) Describe the boundary conditions that must be satisfied by the wavefunctions ψ(x) (such as energy eigenfunctions). (b) Solve the Schr¨odinger’s equation and by using the boundary conditions of part (a) find all energy eigenfunctions, ψn(x), and the corresponding energies, En. (c) What are...

1. Consider a particle of mass m in a box of length L with boundaries at x...

1. Consider a particle of mass m in a box of length L with boundaries at x = 0 and x = L. At t = 0 the wavefunction is where A is the normalization constant. (a) Determine the basis eigen states for a particle in the box. (b) Determine the normalization constant A. (c) Determine the probability of finding the particle in the ground state at t ≠ 0. (d) Show that the sum of probabilities of finding the...

Consider the one-dimensional motion of a particle of mass ? in a space where uniform gravitational...

Consider the one-dimensional motion of a particle of mass ? in a space where uniform gravitational acceleration ? exists. Take the vertical axis ?. Using Heisenberg's equation of motion, find the position-dependent operators ? ? and momentum arithmetic operator ?? in Heisenberg display that depend on time. In addition, calculate ??, ?0, ??, ?0.

Suppose a particle of mass m and charge q is in a one-dimensional harmonic oscillator potential...

Suppose a particle of mass m and charge q is in a one-dimensional harmonic oscillator potential with natural frequency ω0. For times t > 0 a time-dependent potential of the form V1(x) = εxcos(ωt) is turned on. Assume the system starts in an initial state|n>. 1. Find the transition probability from initial state |n> to a state |n'> with n' ≠ n. 2. Find the transition rate (probability per unit time) for the transition |n>→|n'>.

Consider the one dimensional model of one-particle-in-a-box. Under what condition the two quantum levels are orthogonal....

Consider the one dimensional model of one-particle-in-a-box. Under what condition the two quantum levels are orthogonal. Namely, find the relation between m and n so that < m | n > = 0

For a particle of mass m in a two-dimensional potential well(i.e. particle confined in a 2D...

For a particle of mass m in a two-dimensional potential well(i.e. particle confined in a 2D box) with sides L1 and L2: a) Set up the Schrödinger equation for this system. b) Using separation of variables, derive an expression for thewavefunction Ψ and for the energy levels of the system. Be sure toshow all of your work for this derivation. c) For the case of square potential well with side L (i.e.L1=L2=L), sketch an energy level diagram forthe nine lowest...

5. Consider a particle in a two-dimensional, rigid, square box with side a. (a) Find the...

5. Consider a particle in a two-dimensional, rigid, square box with side a. (a) Find the time independent wave function φ(x,y)describing an arbitrary energy eigenstate. (b)What are the energy eigenvalues and the quantum numbers for the three lowest eigenstates? Draw the energy level diagram

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Consider a particle of mass m moving in a two-dimensional harmonic oscillator potential : U(x,y)=...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Consider a particle of mass m moving in a two-dimensional harmonic oscillator potential : U(x,y)= 1/2 mω^2 (x^2+y^2 ) a. Use separation of variables in Cartesian coordinates to solve the Schroedinger equation for this particle. b. Write down the normalized wavefunction and energy for the ground state of this particle. c. What is the energy and degeneracy of each of the lowest 5 energy levels of this particle? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Use the quantum particle wavefunctions for the kinetic energy levels in a one dimensional box to...

Use the quantum particle wavefunctions for the kinetic energy levels in a one dimensional box to qualitatively demonstrate that the classical probability distribution (any value of x is equally allowed) is obtained for particles at high temperatures.

Question