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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Simplify the following Boolean function using K-map:
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second Q. Design the simplified function using iCircuit then submit your
simulation in Canvas as Lab #4
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.
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Apply the boundary conditions to the finite square well potential at x = L to find the following.
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and D.)
Y is continuous
B =
is continuous
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(b) the ratio C/D (Use the following as necessary: a, k, and L.)
Chapter 8 Solutions
Introduction To Quantum Mechanics
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- 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 Enarrow_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
- one-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_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_forwardSuppose a particle confined to a cavity in a microporous material has a potential energy of the form V(x) = V0 (e-a^2/x^2 - 1). Sketch V(x). What is the value of the force constant corresponding to this potential energy? Would the particle undergo simple harmonic motion? Sketch the likely form of the first two vibrational wavefunctions.arrow_forward
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