Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 8.1, Problem 8.1P
(a)
To determine
The linear potential
(b)
To determine
The quartic potential
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a) A particle is placed in the well with an energy E< Uo. Sketch the first three energy levels, and make
sure to label each one.
V(x)
U.
+2(x)
L
X
X
Simplify the following Boolean function using K-map:
f=E (0,1,2,5,7,8,9,10) & don't care X-2 (3,13,14)
ZW
ху
X
W
Solution:
f=
second Q. Design the simplified function using iCircuit then submit your
simulation in Canvas as Lab #4
Consider the three-dimensional harmonic oscillator, for which the potential is
V ( r ) = 1/2 m ω2 r2
(a) Show that the separation of variables in Cartesian coordinates turns this into three one-dimensional oscillators, and exploit your knowledge of the latter to determine the allowed energies.
Answer: En = ( n + 3/2 ) ħ ω
(b) Determine the degeneracy d ( n ) of En
Chapter 8 Solutions
Introduction To Quantum Mechanics
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Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Similar questions
- Apply the boundary conditions to the finite square well potential at x = L to find the following. (a) the relationships among the coefficients B, C, and D due to boundary conditions (Use the following as necessary: a, k, L, C and D.) Y is continuous B = is continuous dx B = (b) the ratio C/D (Use the following as necessary: a, k, and L.)arrow_forwardPROBLEM 3. Using the variational method, calculate the ground state en- ergy Eo of a particle in the triangular potential: U(r) = 0 r 0. Use the trial function v(x) = Cx exp(-ar), where a is a variational parameter and C is a normalization constant to be found. Compare your result for Eo with the exact solution, Eo 1.856(h? F/m)/3.arrow_forwardAn electron with energy E= +4.80 eV is put in an infinite potential well with U(x) =infinity for x<0 and x>L. Of course, U(x) = 0 for 0<x<L. Find the largest amount of time that the electron can exist outside the box. Draw and Label a figure.arrow_forward
- 5:A): Consider a system is in the state = 3 8x . Find the eigenvalue of £, ?arrow_forwardA spin 1/2 system is known to be in an eigenstate of Sn = (Sx+S₂)/√2 with eigenvalue +ħ/2. Find the probability that a measurement of S in this state, will yield the value +ħ/2. Xarrow_forwardA particle moves in one dimension x under the influence of a potential V(x) as sketched in the figure below. The shaded region corresponds to infinite V, i.e., the particle is not allowed to penetrate there. V(x) a b a²Vo = If there is an energy eigenvalue E = 0, then a and V, are related by a²Vo = (n + ² ) ² n² 2m 3-1 n²π² 2m a²V₁ = (n + ²) π ² 2m -Vo nπ² 0 a Xarrow_forward
- 40. The first excited state of the harmonic oscillator has a wave function of the form y(x) = Axe-ax². (a) Follow thearrow_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_forwardFor the function f(z) = 2+2 of complex variable z, which of the following statements is incorrect? z2-2z Select one: Oa. z=0 is a simple pole with residue -1 Ob. z=2 is a simple pole with residue 2 ○ c. Both the first two options are correct O d. None of the first two options are correctarrow_forward
- Problem 4. Consider two indistinguishable, noninteracting spin-1/2 fermions in a one- dimensional infinite square well potential of length L. (a) What is the ground-state energy of the two-particle system? (b) What is the ground-state quantum state vector? (c) What is the first excited state energy of the two-particle system? (d) What are the quantum state vectors of the first excited state? (e) What is the degeneracy of the first excited state? (f) Discuss qualitatively how the excited-state energies change if we consider the particles to be interacting through the Coulomb potential.arrow_forwardPROBLEM 1. Calculate the normalized wave function and the energy level of the ground state (1 = 0) for a particle in the infinite spherical potential well of radius R for which U(r) = 0 at r R. HINT: Reduce the spherically-symmetric SE to a ld form using the substi- tution (r) = x(r)/r.arrow_forwardone-dimensional A one-particle, system has the potential energy function V = V₁ for 0 ≤ x ≤ 1 and V = ∞ elsewhere (where Vo is a constant). a) Use the variation function = sin() for 0 ≤ x ≤ 1 and = 0 elsewhere to estimate the ground-state energy of this system. b) Calculate the % relative error.arrow_forward
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