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 7.1, Problem 7.6P
(a)
To determine
Show that there is no first order change in energy levels and calculate second order correction.
(b)
To determine
The exact energies and show they are consistent with perturbation theory approximation.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
For the questions 4 and 5 consider the following:
A harmonic oscillator where the time-independent Schrodinger equation
for the nth allowed energy is
(2) + m²(2) = Ent(2),
where E-
Question 4
0000
(c) 9,
(4) 4₂
Calculate the standard deviation in position z and momentum p (L.e.,
o, and a) for t
(b)
Question 5
2m dr
+¹1) ₁
n+
149
with neNº= (0,1,2,...).
8pm
Calculate the standard deviation in position z and momentum p (ie,
a, and o,) for ₁.
Problem 1:
(a) A non-relativistic, free particle of mass m is bouncing back and forth between two perfectly reflecting
walls separated by a distance L. Imagine that the two oppositely directed matter waves associated with this
particle interfere to create a standing wave with a node at each of the walls. Find the kinetic energies of the
ground state (first harmonic, n = 1) and first excited state (second harmonic, n = 2). Find the formula for
the kinetic energy of the n-th harmonic.
(b) If an electron and a proton have the same non-relativistic kinetic energy, which particle has the larger
de Broglie wavelength?
(c) Find the de Broglie wavelength of an electron that is accelerated from rest through a small potential
difference V.
(d) If a free electron has a de Broglie wavelength equal to the diameter of Bohr's model of the hydrogen
atom (twice the Bohr radius), how does its kinetic energy compare to the ground-state energy of an electron
bound to a Bohr model hydrogen atom?
Problem 2.34:- Show that E must be greater than minimum value of Ve for every normalizeable solution to time independent Schrodinger wave equation.
Chapter 7 Solutions
Introduction To Quantum Mechanics
Ch. 7.1 - Prob. 7.1PCh. 7.1 - Prob. 7.2PCh. 7.1 - Prob. 7.3PCh. 7.1 - Prob. 7.4PCh. 7.1 - Prob. 7.5PCh. 7.1 - Prob. 7.6PCh. 7.2 - Prob. 7.8PCh. 7.2 - Prob. 7.9PCh. 7.2 - Prob. 7.10PCh. 7.2 - Prob. 7.11P
Ch. 7.2 - Prob. 7.12PCh. 7.2 - Prob. 7.13PCh. 7.3 - Prob. 7.15PCh. 7.3 - Prob. 7.16PCh. 7.3 - Prob. 7.17PCh. 7.3 - Prob. 7.18PCh. 7.3 - Prob. 7.19PCh. 7.3 - Prob. 7.20PCh. 7.3 - Prob. 7.21PCh. 7.3 - Prob. 7.22PCh. 7.4 - Prob. 7.23PCh. 7.4 - Prob. 7.24PCh. 7.4 - Prob. 7.25PCh. 7.4 - Prob. 7.26PCh. 7.4 - Prob. 7.27PCh. 7.4 - Prob. 7.28PCh. 7.4 - Prob. 7.29PCh. 7.5 - Prob. 7.31PCh. 7.5 - Prob. 7.32PCh. 7 - Prob. 7.33PCh. 7 - Prob. 7.34PCh. 7 - Prob. 7.35PCh. 7 - Prob. 7.36PCh. 7 - Prob. 7.37PCh. 7 - Prob. 7.38PCh. 7 - Prob. 7.39PCh. 7 - Prob. 7.40PCh. 7 - Prob. 7.42PCh. 7 - Prob. 7.43PCh. 7 - Prob. 7.44PCh. 7 - Prob. 7.45PCh. 7 - Prob. 7.46PCh. 7 - Prob. 7.47PCh. 7 - Prob. 7.49PCh. 7 - Prob. 7.50PCh. 7 - Prob. 7.51PCh. 7 - Prob. 7.52PCh. 7 - Prob. 7.54PCh. 7 - Prob. 7.56PCh. 7 - Prob. 7.57P
Knowledge Booster
Similar questions
- PROBLEM 2. Consider a spherical potential well of radius R and depth Uo, so that the potential is U(r) = -Uo at r R. Calculate the minimum value of Uc for which the well can trap a particle with l = 0. This means that SE at Uo > Uc has at least one bound ground state at l = 0 and E < 0. At Ug = Uc the bound state disappears.arrow_forwardSuppose a harmonic oscillator is subject to a perturbation av = Ahw (&/#0)* . where ro = mw/h is the length scale of the problem. a) Use Rayleigh-Schrödinger perturbation theory to find the first and second order corrections to the energies of the n'th level. b) Discuss the applicability of the perturbative approach for states with large n,arrow_forward40. The first excited state of the harmonic oscillator has a wave function of the form y(x) = Axe-ax². (a) Follow thearrow_forward
- PROBLEM 3. Using the variational method, calculate the ground state en- ergy Eo of a particle in the triangular potential: U(r) = 0 r 0. Use the trial function v(x) = Cx exp(-ar), where a is a variational parameter and C is a normalization constant to be found. Compare your result for Eo with the exact solution, Eo 1.856(h? F/m)/3.arrow_forwardLet'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_forward2.4. A particle moves in an infinite cubic potential well described by: V (x1, x2) = {00 12= if 0 ≤ x1, x2 a otherwise 1/2(+1) (a) Write down the exact energy and wave-function of the ground state. (2) (b) Write down the exact energy and wavefunction of the first excited states and specify their degeneracies. Now add the following perturbation to the infinite cubic well: H' = 18(x₁-x2) (c) Calculate the ground state energy to the first order correction. (5) (d) Calculate the energy of the first order correction to the first excited degenerated state. (3) (e) Calculate the energy of the first order correction to the second non-degenerate excited state. (3) (f) Use degenerate perturbation theory to determine the first-order correction to the two initially degenerate eigenvalues (energies). (3)arrow_forward
- 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-->)arrow_forwardStarting with the equation of motion of a three-dimensional isotropic harmonic ocillator dp. = -kr, dt (i = 1,2,3), deduce the conservation equation dA = 0, dt where 1 P.P, +kr,r,. 2m (Note that we will use the notations r,, r2, r, and a, y, z interchangeably, and similarly for the components of p.)arrow_forwardA proton is confined in box whose width is d = 750 nm. It is in the n = 3 energy state. What is the probability that the proton will be found within a distance of d/n from one of the walls? Include a sketch of U(x) and ?(x). Sketch the situation, defining all your variablesarrow_forward
- Problem 2. Consider the double delta-function potential V(x) = a[8(x + a) + 8(x − a)], where a and a are positive constants. (a) Sketch this potential. (b) How many bound states does it possess? Find the allowed energies, for a = ħ²/ma and for a = ħ²/4ma, and sketch the wave functions.arrow_forwardConsider the Schrodinger equation for a one-dimensional linear harmonic oscillator: -(hbar2/2m) * d2ψ/dx2 + (kx2/2)*ψ(x) = Eψ(x) Substitute the wavefunction ψ(x) = e-(x^2)/(ξ^2) and find ξ and E required to satisfy the Schrodinger equation. [Hint: First calculate the second derivative of ψ(x), then substitute ψ(x) and ψ′′(x). After this substitution, there will be an overall factor of e-(x^2)/(ξ^2) on both sides of the equation which canbe an canceled out. Then, gather all terms which depend on x into one coefficient multiplying x2. This coefficient must be zero because the equation must be satisfied for any x, and equating it with zero yields the expression for ξ. Finally, the remaining x-independent part of the equation determines the eigenvalue for energy E associated with this solution.]arrow_forward6.2. Solve the three-dimensional harmonic oscillator for which 1 V(r) = -— mw² (x² + y²² +2²) 2 by separation of variables in Cartesian coordinates. Assume that the one-dimensional oscillator has eigenfunctions (x) with corresponding energy eigenvalues En = (n + 1/2)hw. What is the degeneracy of the first excited state of the oscillator?arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage Learning
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning