Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 2.6, Problem 2.31P
To determine
The bound state energy for the delta-function potential, by treating it as the limit of a finite square well and to check its consistency with
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Chapter 2 Solutions
Introduction To Quantum Mechanics
Ch. 2.1 - Prob. 2.1PCh. 2.1 - Prob. 2.2PCh. 2.2 - Prob. 2.3PCh. 2.2 - Prob. 2.4PCh. 2.2 - Prob. 2.5PCh. 2.2 - Prob. 2.6PCh. 2.2 - Prob. 2.7PCh. 2.2 - Prob. 2.8PCh. 2.2 - Prob. 2.9PCh. 2.3 - Prob. 2.10P
Ch. 2.3 - Prob. 2.11PCh. 2.3 - Prob. 2.12PCh. 2.3 - Prob. 2.13PCh. 2.3 - Prob. 2.14PCh. 2.3 - Prob. 2.15PCh. 2.3 - Prob. 2.16PCh. 2.4 - Prob. 2.17PCh. 2.4 - Prob. 2.18PCh. 2.4 - Prob. 2.19PCh. 2.4 - Prob. 2.20PCh. 2.4 - Prob. 2.21PCh. 2.5 - Prob. 2.22PCh. 2.5 - Prob. 2.23PCh. 2.5 - Prob. 2.24PCh. 2.5 - Prob. 2.25PCh. 2.5 - Prob. 2.26PCh. 2.5 - Prob. 2.27PCh. 2.5 - Prob. 2.28PCh. 2.6 - Prob. 2.29PCh. 2.6 - Prob. 2.30PCh. 2.6 - Prob. 2.31PCh. 2.6 - Prob. 2.32PCh. 2.6 - Prob. 2.34PCh. 2.6 - Prob. 2.35PCh. 2 - Prob. 2.36PCh. 2 - Prob. 2.37PCh. 2 - Prob. 2.38PCh. 2 - Prob. 2.39PCh. 2 - Prob. 2.40PCh. 2 - Prob. 2.41PCh. 2 - Prob. 2.42PCh. 2 - Prob. 2.44PCh. 2 - Prob. 2.45PCh. 2 - Prob. 2.46PCh. 2 - Prob. 2.47PCh. 2 - Prob. 2.49PCh. 2 - Prob. 2.50PCh. 2 - Prob. 2.51PCh. 2 - Prob. 2.52PCh. 2 - Prob. 2.53PCh. 2 - Prob. 2.54PCh. 2 - Prob. 2.58PCh. 2 - Prob. 2.63PCh. 2 - Prob. 2.64P
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- Problem 2.3 Show that there is no acceptable solution to the (time-independent) Schrödinger equation (for the infinite square well) with E = 0 or E < 0. (This is a special case of the general theorem in Problem 2.2, but this time do it by explicitly solving the Schrödinger equation and showing that you cannot meet the boundary conditions.)arrow_forwardCompute the vector potential A for z > 0 in the Coulomb gauge corresponding to the current J = K8(z)ÿ if Az = Ax = 0 and %3D lim, 0+ Ay = 0. Integrate the Poisson's equation analog to obtain your answer.arrow_forwardProblem #1 (Problem 5.3 in book). Come up with a function for A (the Helmholtz free energy) and derive the differential form that reveals A as a potential: dA < -SdT – pdV [Eqn 5.20]arrow_forward
- A point particle moves in space under the influence of a force derivablefrom a generalized potential of the formU(r, v) = V (r) + σ · L,where r is the radius vector from a fixed point, L is the angular momentumabout that point, and σ is the fixed vector in space. Find the components of the force on the particle in spherical polar coordinates, on the basis of the equation for the components of the generalized force Qj: Qj = −∂U/∂qj + d/dt (∂U/∂q˙j)arrow_forwardProblem 2.3 Show that there is no acceptable solution to the (time-independent) Schrödinger equation for the infinite square well with E = 0 or E < 0. (This is a special case of the general theorem in Problem 2.2, but this time do it by explicitly solving the Schrödinger equation, and showing that you cannot satisfy the boundary conditions.)arrow_forwardQuestion 2 2.1 Consider an infinite well for which the bottom is not flat, as sketched here. If the slope is small, the potential V = 6 |x|/ a may be considered as a perturbation on the square- well potential over -a/2 ≤x≤a/2. -8 W V(x) a/2 -a/2 X Calculate the ground-state energy, correct to first order in perturbation theory. Given (0) = √²/co COS Ground state of box of size a: = Ground state energy: E(0) = 4²k² 2ma². 0 Yarrow_forward
- Consider a system of five particles, inside a container where the allowed energy levels are nondegenerate and evenly spaced. For instance, the particles could be trapped in a one-dimensional harmonic oscillator potential. In this problem you will consider the allowed states for this system, depending on whether the particles are identical fermions, identical bosons, or distinguishable particles. Describe the ground state of this system, for each of these three cases.arrow_forwardA point particle moves in space under the influence of a force derivablefrom a generalized potential of the formU(r, v) = V (r) + σ · L,where r is the radius vector from a fixed point, L is the angular momentumabout that point, and σ is the fixed vector in space. Find the components of the force on the particle in Cartesian coordinates, on the basis of the equation for the components of the generalized force Qj: Qj = −∂U/∂qj + d/dt (∂U/∂q˙j)arrow_forwardIn this question we will consider a finite potential well in which V = −V0 in the interval −L/2 ≤ x ≤ L/2, and V = 0 everywhere else (where V0 is a positive real number). For a particle with in the range −V0 < E < 0, write and solve the time-independent Schrodinger equation in the classically allowed and classically forbidden regions. Remember to keep the wavenumbers and exponential factors in your solutions real!arrow_forward
- Consider a two-dimensional infinite rectangular well, with a potential given by V (x, y) = { 0 0 ≤ x ≤ a; 0 ≤ y ≤b otherwise Using separation of variables in cartesian coordinates (as worked out in class for the 3-dimensional case) :{% Find the stationary states and corresponding energy eigenvalues (b) Write down the first 5 distinct energy eigenstates for the case b = a. How would your answer to (b) above change if b = 2a? (d) Write down the first 5 distinct energy eigenstates for the case b » a.arrow_forward(a_)*(a_V)ax- J-o Problem 2.11 Show that the lowering operator cannot generate a state of infinite norm (i.e., f la.-²dx < oo, if y itself is a normalized solution to the Schrödinger equation). What does this tell you in the case y = vo? Hint: Use integration by parts to show that y*(a,a_) dx. = -00 Then invoke the Schrödinger equation (Equation 2.46) to obtain la-yl² dx E - hw, -0- where E is the energy of the state y. **Problem 2.12 (a) The raising and lowering operators generate new solutions to the Schrödinger equation, but these new solutions are not correctly normalized. Thus a Vn is proportional to yn+1, and a n is proportional to yn-1, but we'd like to know the precise proportionality constants. Use integration by parts and the Schrödinger equation (Equations 2.43 and 2.46) to show that roo | la+ Vl² dx = (n+ 1)hw, la- Vnl? dx = nhw, -00 -00 and hence (with i's to keep the wavefunctions real) a+ Vn = iv(n + 1)hw yn+1, [2.52] a_n = -ivnhw n-1. [2.53] Sec. 2.3: The Harmonic…arrow_forwardfor a particle inside a finite well its parity can either be even or odd. if the well is symmetric about its center then it has even parity as i understand. So for straight finite potential barriers, no matter the length or if you change its position, a well that has straight, finite , parallel boundary walls will always be symmetric then, is this correct?arrow_forward
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