Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 9.1, Problem 9.1P
To determine
The allowed energies
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Question 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
Y
where
and
kyk₂
I
2
k
2m E
2
ħ²
2m
ħ²
(V-E)
3 Show that the solutions for region II
can also be written as
2/₁₂ (²) = Ccas (₁₂²) + D sin (k₂²)
for Z≤ 1W/
4 Since the potential well Vez) is symmetrical,
the possible eigen functions In You will
be symmetrical, so Yn will be either
even or odd.
a) write down the even solution for
region. II
b) write down the odd solution
region
for
on II
In problem 2, explain why A=G=0₁
Q.8: Find the center and radius of convergence of the power series 2n=1 m(n+1)
(z)2n+1
Q.9: Find the McLaurin series and its radius of convergence of
3iz
Q.10: Find the Laurent series that converges for 0 < I z – z,I < R and determine the precise
z, = iT
region of convergence:
(z-in)*
Chapter 9 Solutions
Introduction To Quantum Mechanics
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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_forwardPROVE: Triangle Inequality We prove Property 5 of abso- lute values, the Triangle Inequality: |x + y < |x| + y| (a) Verify that the Triangle Inequality holds for x = 2 and y = 3, for x = -2 and y = -3, and for x = -2 and y = 3. (b) Prove that the Triangle Inequality is true for all real num- bers x and y. [Hint: Take cases.]arrow_forwardThe potential energy within the embedded atom method (EAM) formalism is ex- pressed as (5) A special case of EAM potentials is the Finnis-Sinclair (FS) type of potential where -√√₁. Derive the pairwise forces within a dimer (between two isolated atoms), separated by the distance, ro, in for FS potentials. F(pi) = U = [Fi(P₁) + = ΣV(rij), Pi = = [ f(rij) jfiarrow_forward
- Calculate the values of ⟨x3⟩v and ⟨x4⟩v for a harmonic oscillator by using the properties of the Hermite polynomials given in Table 7E.1; follow the approach used in the text.arrow_forwardGiven the function below, determine if the function is continuous at the point I why. 3. If not, indicate | -x - 3, r < - 3 f(x) = 4, 3D3 - T + 3, -3 OContinuous at I - 3 ONot continuous: f( - 3) does not exist ONot continuous: lim f(x) does not exist ONot continuous: f(-3) and limit exist, but are not equalarrow_forwardFind A and B so that f(x, y) = x2 + Ax + y? + B has a local minimum at the point (4, 0), with z-coordinate 5. A = B =arrow_forward
- Consider a particle confined to an infinite square potential well with walls at x = 0 and x = L. Calculate the probability of finding the particle in the n" energy eigenstate and somewhere in the first (1/a) th of the well (ie. in the region starting from one side of the well and extending L/a into the well).arrow_forwardFind the Laurent series for the following functions about the indicated points; hence find the residue of the function at the point. (Be sure you have the Laurent series which converges near the point.) 2. z(z – 1) z = 1arrow_forwardFind the Taylor polynomials of orders n = 0, 1, 2, 3, and 4 about x = xo and then find the nth Taylor polynomial for the function in sigma notation. 5 In(x); xo = e po(x) = pi(x) = P2(x) = P3(x) = P4(x) = Σ k =arrow_forward
- Find the poisson brackets [F,G] for the functions: F = q1 + q2pỉ and G = q + 2p 3 Answer:arrow_forwardUse a trial function of the form e(-ax^2)/2 to calculate the ground state energy of a quartic oscillator, whose potential is V(x)=cx4. Show full procedure in a clear way. Do not skip any stepsarrow_forwardThe Brachistochrone Problem: Show that if the particle is projected withan initial kinetic energy 1/2 m v02 that the brachistochrone is still a cycloidpassing through the two points with a cusp at a height z above the initialpoint given by v02 = 2gz.arrow_forward
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