Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
expand_more
expand_more
format_list_bulleted
Question
Chapter 9.3, Problem 9.6P
(a)
To determine
The potential energy above the floor in terms of
(b)
To determine
To solve the Schrodinger equation having potential
(c)
To determine
The first four allowed energies in joules.
(d)
To determine
The ground state energy of electron in
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Consider the half oscillator" in which a particle of mass m is restricted to the region x > 0 by the potential energy
U(x) = 00
for a O
where k is the spring constant.
What are the energies of the ground state and fırst excited state? Explain your reasoning.
Give the energies in terms of the oscillator frequency wo =
Vk/m.
Formulas.pdf (Click here-->)
Q # 01: Consider a ball of 100g dropped with zero velocity from the height of 2m.
Estimate its total energy in eV units.
Using the energy conservation, find out the velocity as a function of position of the ball. Sketch its phase trajectory.
Calculate the time it takes to reach the ground.
Let’s assume that it is bounced back with no loss in its total energy.Will it reachthe same height? Make an analytical argument.
What if the collision with the ground is not elastic and it loses some of its energy (which energy?).The ball will eventually come to rest after bouncing few times. Sketch the phase trajectory for the whole duration.
What is the range of total energy of this system? Can the energy of this systemassume discrete values? Explain mathematically.
Consider the three-dimensional harmonic oscillator, for which the potential is
V ( r ) = 1/2 m ω2 r2
(a) Show that the separation of variables in Cartesian coordinates turns this into three one-dimensional oscillators, and exploit your knowledge of the latter to determine the allowed energies.
Answer: En = ( n + 3/2 ) ħ ω
(b) Determine the degeneracy d ( n ) of En
Chapter 9 Solutions
Introduction To Quantum Mechanics
Knowledge Booster
Similar questions
- Let's consider a harmonic oscillator. The total energy of this oscillator is given by E=(p²/2m) +(½)kx?. A) For constant energy E, graph the energies in the range E to E + dE, the allowed region in the classical phase space (p-x plane) of the oscillator. B) For k = 6.0 N / m, m = 3.0 kg and the maximum amplitude of the oscillator xmax =2.3 m For the region with energies equal to or less than E, the oscillator number of states that can be entered D(E).arrow_forwardWrite down the equations and the associated boundary conditions for solving particle in a 1-D box of dimension L with a finite potential well, i.e., the potential energy U is zero inside the box, but finite outside the box. Specifically, U = U₁ for x L. Assuming that particle's energy E is less than U, what form do the solutions take? Without solving the problem (feel free to give it a try though), qualitatively compare with the case with infinitely hard walls by sketching the differences in wave functions and probability densities and describing the changes in particle momenta and energy levels (e.g., increasing or decreasing and why), for a given quantum number.arrow_forwardPhysics Department PHYS4101 (Quantum Mechanics) Assignment 2 (Fall 2020) Name & ID#. A three-dimensional harmonic oscillator of mass m has the potential energy 1 1 1 V(x.y.2) = ; mw*x² +mwży² +=mw;z? where w1 = 2w a. Write its general eigenvalues and eigenfunctions b. Determine the eigenvalues and their degeneracies up to the 4th excited state c. The oscillator is initially equally likely found in the ground, first and second excited states and is also equally likely found among the states of the degenerate levels. Calculate the expectation values of the product xyz at time tarrow_forward
- Use a trial function of the form e(-ax^2)/2 to calculate the ground state energy of a quartic oscillator, whose potential is V(x)=cx4. Show full procedure in a clear way. Do not skip any stepsarrow_forwardConsider the particle in the infinite potential well as shown in Figure P2.29. Derive and sketch the wave functions corresponding to the four lowest energy levels. (Do not normalize the wave functions.)arrow_forwardA particle with mass m is moving along the x-axis in a potential given by the potential energy function U(x) = 0.5mw²x². Compute the product (x, t)*U (x) V (x, t). Express your answer in terms of the time-independent wave function, (x).arrow_forward
- 5.1 Consider a one-dimensional bound particle. Show that if the particle is in a stationary state at a given time, then it will always remain in a stationary state. 5.2 Consider a one-dimensional bound particlearrow_forwardInfinite/finite Potential Well 1. Sketch the solution (Wave function - Y) for the infinite potential well and show the following: (a) Specify the boundary conditions for region I, region II, and region III. (i.e. U = ?, and x = ?) (b) Specify the length of the potential well (L=10 cm) (c) Which region will have the highest probability of finding the particle?arrow_forwardConsider the one-dimensional time-independent Schrödinger equation for some arbitrary potential V(x). Prove that if a solution p(x) has the property that (x) → 0 as r → ±00, then the solution must be nondegen- erate and therefore real, apart from a possible overall phase factor. Hint: Show that the contrary assumption leads to a contradiction.arrow_forward
- 40. The first excited state of the harmonic oscillator has a wave function of the form y(x) = Axe-ax². (a) Follow thearrow_forwardConsider a particle of mass, m, with energy, E, moving to the right from -o. This particle is (0 x V. xz a Sketch a picture that shows the potential energy. In this picture represent a particle moving to the right when x < 0. Solve the time independent Schrodinger equation to find E(x) on the domain -oarrow_forwardConsider a 1-dimensional quantum system of one particle Question 01: in which the particle is under a potential V(x) = mw?a?, with m being the mass of the particle and w being a parameter (you may take it as angular fre- quency) with inverse dimension of time. The particle may be found in the region -0 < x < o. Varify that the lowest two states of the system are mutually orthonormal.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
Recommended textbooks for you
Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage Learning
Modern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage Learning
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning
Modern Physics
Physics
ISBN:9781111794378
Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher:Cengage Learning