Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 1.4, Problem 1.5P
(a)
To determine
The normalized wave function.
(b)
To determine
The expectation value of
(c)
To determine
The standard deviation of
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Given that at time t = 0 a particle’s wave function is given by ψ(x, 0) =Ax/a, if 0 ≤ x ≤ a,A(b − x)/(b − a), if a ≤ x ≤ b, with A0, Otherwise.a and b as constants, answer the following questions;
a) Find the normalization constant A in terms of the constants a and b.
b) Sketch ψ(x, 0) as a function of x.
c) Where is the particle most likely to be found at time t = 0?
d) What is the probability of finding the particle to the left of a?
Consider the wave function for the ground state harmonic oscillator:
m w1/4
e-m w x2/(2 h)
A. What is the quantum number for this ground state? v = 0
B. Enter the integrand you'd need to evaluate (x) for the ground state harmonic oscillator wave 'function: (x) = |-
то
dx
e
C. Evaluate the integral in part B. What do you obtain for the average displacement? 0
Consider 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.]
Chapter 1 Solutions
Introduction To Quantum Mechanics
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- Suppose 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_forwardThe uncertainties of a position and a momentum of a particle (Ax) and (Ap) are defined as Ax = /(x²) – (x)² Ap = /(p*) – {p}² 1. For the particle in the box at the ground eigenstate (n = 1) and first excited state ( 2), what is the uncertainty in the value x? How would you interpret the results of these calculations? n = 2. For the particle in the box at the ground eigenstate (n = 1) and first excited state ( n = 2), what is the uncertainty in the value p? How would you interpret the results of these calculations? 3. What is the product for the ground and first excited state: AxAp. 4. Does the Heisenberg Uncertainty Principle hold for a particle in each of these states?arrow_forward2i+1 i+1 |- +> + 3 [recall, |+ -> means that particle #1 is in the |+> state (usual Z basis) and #2 is in the |-> state.] A) Show that this state is already normalized. B) Is this state separable or entangled? C) A measurement of S, is made on particle #1. What are the possible results and with what probabilities? D) A measurement of Sz is made on particle #2. What are the possible results and with what probabilities? E) Calculate the expectation value of the correlation function between these two measurements . (Don't use matrices -- use probabilities!)arrow_forward
- The eigenstates of the particle-in-a-box are written, n = √ sin (™T). If L = 10.0, what is the expectation value for the quantity 2ħ² + p² in the n = 3 eigenstate? Report your answer as a multiple of ħ². (Note: ô = −iħª) d dx'arrow_forwardConsider the function v(1,2) =( [1s(1) 3s(2) + 3s(1) 1s(2)] [x(1) B(2) + B(1) a(2)] Which of the following statements is incorrect concerning p(1,2) ? a. W(1,2) is normalized. Ob. The function W(1,2) is symmetric with respect to the exchange of the space and the spin coordinates of the two electrons. OC. y(1,2) is an eigenfunction of the reference (or zero-order) Hamiltonian (in which the electron-electron repulsion term is ignored) of Li with eigenvalue = -5 hartree. d. The function y(1,2) is an acceptable wave function to describe the properties of one of the excited states of Lit. Oe. The function 4(1,2) is an eigenfunction of the operator S,(1,2) = S;(1) + S,(2) with eigenvalue zero.arrow_forwardProblem 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_forward
- Physics 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_forwardFor a particle in a 1-dimensional infinitely deep box of length L, the normalized wave function or the 1st excited state can be written as: Ψ2(x) = {1/i(2L)1/2} ( eibx -e-ibx), where b = 2π/L. Give the full expression that you need to solve to determine the probalibity of finding the particle in the 1st third of the box. Simplify as much as possible but do not solve any integrals.arrow_forwardLet a⪯b⪯c⪯da⪯b⪯c⪯d be the variable ordering.ϕ=ϕ= a&b&d&!c|a&c&d|d&!b&!c|!dβ=β= a&b&c|!c a) Convert the formula ϕϕ to Shannon normal form. b) Convert the formula ββ to Shannon normal form. c) ψψ is obtained by replacing all occurences of the variable b by formula ββ in formula ϕϕ.Compute the ROBDD of ψψ by the Compose algorithm, and convert the result to Shannon normal form.arrow_forward
- 3. Plane waves and wave packets. In class, we solved the Schrodinger equation for a "free particle" (e.g. when U(x,t) = 0). The correct[solution is (x, t) = Ae(px-Et)/ħ This represents a "plane wave" that exists for all x. However, there is a strange problem with this: if you try to normalize the wave function (determine A by integrating * for all x), you will find an inconsistency (A has to be set equal to 0?). This is because the plane wave stretches to infinity. In order to actually represent a free particle, this solution needs to be handled carefully. Explain in words (and/or diagrams) how we can construct a "wave packet" from the plane wave solution. (Hint 1: consider a superposition of plane waves for a limited range of momentum/energy. Hint 2: have a look at the brief discussion in the middle of pg. 278 and especially pg. 308-309 of the text.)arrow_forwardAt time t = 0 the wave function for a particle in a box is given by the function in the provided image, where ψ1(x) and ψ1(x) are the ground-state and first-excited-state wave functions with corresponding energies E1 and E2, respectively. What is ψ(x, t)? What is the probability that a measurement of the energy yields the value E1? What is <E>?arrow_forwardPhysics Use qualitative arguments based on the equation of Schrödinger to sketch wave functions in states with energies E1 < Vd and E2 > Vd at the potential shown in the figure below. Detail: in regions x < 0 and x > b the potential V(x) is very large. Justify your answers in detail for each region.arrow_forward
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