Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Question
Chapter 5.2, Problem 5.12P
(a)
To determine
A solution to a completely symmetric function and a completely antisymmetric function which also satisfy the Schrodinger equation with same energy and determine what happens to the completely antisymmetric function if
(b)
To determine
Show that a completely antisymmetric spin state for
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Students have asked these similar questions
Solve the time-independent Schrödinger equation with appropriate
boundary conditions for an infinite square well centered at the origin [V (x) = 0, for
-a/2 < x < +a/2; V (x) = 00 otherwise]. Check that your allowed energies are
consistent with mine (Equation 2.23), and confirm that your y's can be obtained from
mine (Equation 2.24) by the substitution x x - a/2.
Problem #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]
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)
Chapter 5 Solutions
Introduction To Quantum Mechanics
Ch. 5.1 - Prob. 5.1PCh. 5.1 - Prob. 5.2PCh. 5.1 - Prob. 5.3PCh. 5.1 - Prob. 5.4PCh. 5.1 - Prob. 5.5PCh. 5.1 - Prob. 5.6PCh. 5.1 - Prob. 5.8PCh. 5.1 - Prob. 5.9PCh. 5.1 - Prob. 5.10PCh. 5.1 - Prob. 5.11P
Ch. 5.2 - Prob. 5.12PCh. 5.2 - Prob. 5.13PCh. 5.2 - Prob. 5.14PCh. 5.2 - Prob. 5.15PCh. 5.2 - Prob. 5.16PCh. 5.2 - Prob. 5.17PCh. 5.2 - Prob. 5.18PCh. 5.2 - Prob. 5.19PCh. 5.3 - Prob. 5.20PCh. 5.3 - Prob. 5.21PCh. 5.3 - Prob. 5.22PCh. 5.3 - Prob. 5.23PCh. 5.3 - Prob. 5.24PCh. 5.3 - Prob. 5.25PCh. 5.3 - Prob. 5.26PCh. 5.3 - Prob. 5.27PCh. 5 - Prob. 5.29PCh. 5 - Prob. 5.30PCh. 5 - Prob. 5.31PCh. 5 - Prob. 5.32PCh. 5 - Prob. 5.33PCh. 5 - Prob. 5.34PCh. 5 - Prob. 5.35PCh. 5 - Prob. 5.36PCh. 5 - Prob. 5.38PCh. 5 - Prob. 5.39P
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- 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 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_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_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 meet the boundary conditions.)arrow_forward
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