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 7.4P
(a)
To determine
The exact energies for this two level system.
(b)
To determine
To expand the results obtained in subpart (a) to second order in
(c)
To determine
To show that series in subpart (b) converges only if
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
Given the Hamiltonian:
H=p²/2m+ V(x)
What is the value of the commutator [H,x]?
The Hamiltonian matrix has been constructed using an orthonormal basis.
(1 1 0V
(1 0 1)
A = (2 1 0 )+(0 2 2
\2 1 4
where H = Hº + V and cis a constant.
1 2 0/
b) Use time-independent perturbation theory to determine the eigenvalues with corrections up to second order.
Consider a Hamiltonian of the form
(
H
with a value of << 1.
—
-
a
10 ) + ^ ( 2
Eo
0-Eo
B
Using the perturbation theory, determine the
energy of the system until the correction!
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
- The Hamiltonian of a particle having mass m in one dimension is described by p 1 Н- +mox +2µx. What is the difference between the energies of the first two 2m 2 levels? 2µ? (а) ћо— mo? (b) ћо+ и (с) ћо (d) йо+ moarrow_forwardThe Hamiltonian of a particle having mass m in one dimension is described by = H 2m 2 p²1, +÷mox² +2µx. What is the difference between the energies of the first two levels? 2µ? (а) ћо- mo? (b) ħo+µ (с) ћо (d) ħo+.arrow_forwardShow that the Hamiltonian H = (p²/2m) + V commutes with all three components of L, provided that V depends only on r. (Thus H, L², and L. are mutually compatible observables.)arrow_forward
- The Hamiltonian of a three-level system is represented by the matrix Vo 21 H = 0 2Vo + 1 22 3V. where Vo and A are constants with units of energy (A<< Vo). The correction to the energy level E1=Vo (to second order in A) is: -21/V O 212/Vo O 21/Vo o -212/Vo zeroarrow_forwardThe Hamiltonian of a particle having mass m in one dimension is described by P +-mox? +2µx. What is the difference between the energies of the first two p H 2m 2 levels? 2µ? (а) йо- mo? (b) ħo+ µ (с) ћо (d) ħo+- тоarrow_forwardConsider the Laplace's equation u t uyy = 0 in the square 00 find the associated eigenfunctions X,(x) forn= 1,2,3, - c) Using the boundary condition calculate Y, y) d) Calculate the coefficients (c,) to satisfy the nonhomogeneous condition e) Write a formal series solution of the problem.arrow_forward
- The Hamiltonian of a certain system is given by [1 0 H = ħw]0 LO 0 1 Two other observables A and B are represented by 1 0 0 , B = b]0 2 0 lo o 0- i 0 A = a|-i 0 0 0 1 w, a, b are positive constant. a. Find the eigenvalues and normalized eigenvectors of H b. Suppose the system is initially in the state 2c lµ(0) >= -c 2c where c is a real constant. Determine the normalized state |(t) >. c. What are the eigenvectors of B? d. Find the expectation values of A and B in the state |p(t) >, and hence determine if A and B are conservative observablesarrow_forwardConsider a system of two particles, one with spin 1 and the other with spin 1/2. The particles interact with one another via the Hamiltonian H = Eo [S(1). S(2)] + E₁ [ S:(1) + S. (2)] ħ " where Eo and E₁ are constants with units of energy, S is the spin operator and the labels (1) and (2) indicate the particle on which it operates. (a) Find the energy spectrum of this system, i.e. the eigenvalues of the Hamiltonian H and identify the ground state and first excited state. For numerical purposes assume Eo = 0.2eV and E₁ = 0.1eV. (b) If the two particle system is known to be in a configuration such that the total spin is 1/2, what values a mea- surement of S% of the spin 1 particle would produce and what are the probabilities associated with each one of those values?arrow_forwardThe Hamiltonian of a three-level system is represented by the matrix 22 Vo 2V + 1 22 H = 3V where Vo and A are constants with units of energy (A<< Vo). The corrected eigenstate of the energy level E3=3Vo (to first order in A is:arrow_forward
- Given a Hamiltonian, find eigenvalues and eigenvector Н = 2 (₂² 4 2) 16 2/arrow_forwardSuppose that you have the Lagrangian L = (;2 + ʻr²) + 20 for a 2D 20 system in plane polar coordinates (r, 0). Determine the Hamiltonian.arrow_forwardThe Hamiltonian of a particle having mass m in one dimension is described by p² 1 -+mox² +2ux. What is the difference between the energies of the first two 2m 2 levels?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