Question

Transcribed Image Text:Consider a particle in a box of length L with one end coinciding with the origin. Compute the time-dependent uncertainty in position and
momentum for a particle that is in an equal probability superposition of the n =1 and n = 3 states. Demonstrate the Heisenberg uncertainty
relations in this case.
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by stepSolved in 2 steps

Knowledge Booster
Similar questions
- Solve the time-independent Schrödinger equation and determine the energy levels and the wave function of a particle in the potential a? V (x) = Vol a + 2r2 with a = const.arrow_forwardFor a quantum particle in a scattering state as it interacts a certain potential, the general expressions for the transmission and reflection coefficients are given by T = Jtrans Jinc R = | Jref Jinc (1) where Jinc, Jref, Jtrans are probability currents corresponding to the incident, reflected, and transmitted plane waves, respectively. (a). potential For the particle incident from the left to the symmetric finite square well -Vo; a < x < a, V(x) = 0 ; elsewhere, show that B Ꭲ ; R = A Aarrow_forwardConsider a particle of mass m subject to an attractive delta potential V(x) = - Vod(x), where Vo> 0 (Vo has the dimensions of Energyx Distance). (a) In the case of negative energies, show that this particle has only one bound state; find the binding energy and the wave function. (b) Calculate the probability of finding the particle in the interval -a ≤ x ≤ a. (c) What is the probability that the particle remains bound when Vo is (1) halved suddenly. (ii) quadrupled suddenly? (d) Study the scattering case (i.e., E > 0) and calculate the reflection and transmission coefficients as a function of the wave number k.arrow_forward
- An electron is trapped in a region between two infinitely high energy barriers. In the region between the barriers the potential energy of the electron is zero. The normalized wave function of the electron in the region between the walls is ψ(x) = Asin(bx), where A=0.5nm1/2 and b=1.18nm-1. What is the probability to find the electron between x = 0.99nm and x = 1.01nm.arrow_forwarda question of quantum mechanics: Consider a particle in a two-dimensional potential as shown in the picture Suppose the particle is in the ground state. If we measure the position of the particle, what isthe probability of detecting it in region 0<=x,y<=L/2 ?arrow_forwardConsider a particle moving in a one-dimensional box with walls at x = -L/2 and L/2. (a) Write the wavefunction and probability density for the state n=1. (b) If the particle has a potential barrier at x =0 to x = L/4 (where L = 10 angstroms) with a height of 10.0 eV, what would be the transmission probability of the electrons at the n = 1 state? (c) Compare the energy of the particle at the n= 1 state to the energy of the oscillator at its first excited state.arrow_forward
- Consider a particle in the n = 1 state in a one-dimensional box of length a and infinite potential at the walls where the normalized wave function is given by 2 nTX a y(x) = sin (a) Calculate the probability for finding the particle between 2 and a. (Hint: It might help if you draw a picture of the box and sketch the probability density.)arrow_forwardThe following Eigen function is a typical solution of the time-independent Schrödinger equation and satisfies boundary conditions for a particle in a confined space of a certain length. y(x) = sin (~77) (a) Plot the wave function as a function of x for L = 30 cm and n = 1, 2, 3 and 4. Note: You will need to have 4 plots in the same graph. (b) On a separate graph, plot the probability density (112) as a function of x using the conditions specified in part (a). Note: You will need to have 4 plots in the same graph. (c) Report your observations for parts (a) and (b)arrow_forwardA particle with the velocity v and the probability current density J is incident from the left on a potential step of height Uo, that is, U (x) = Uo at r > 0 and U(x) = 0 at r 0.arrow_forward
- At time t = 0 the normalized wave function for a particle of mass m in the one-dimensional infinite well (see first image) is given by the function in the second image. Find ψ(x, t). What is the probability that a measurement of the energy at time t will yield the result ħ2π2/2mL2? Find <E> for the particle at time t. (Hint: <E> can be obtained by inspection, without an integral)arrow_forwardIn the lab you make a simple harmonic oscillator with a 0.15-kg mass attached to a 12-N/m spring. (a) If the oscillation amplitude is 0.10 m, what is the corresponding quantum number n for the quantum harmonic oscillator? (b) What would be the amplitude of the quantum ground state for this oscillator? (c) What is the energy of a photon emitted when this oscillator makes a transition between adjacent energy levels? Comment on each of your results.arrow_forward
arrow_back_ios
arrow_forward_ios