Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
expand_more
expand_more
format_list_bulleted
Question
Chapter 4.3, Problem 4.23P
(a)
To determine
To prove that
(b)
To determine
To show that
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Question 2: The Rigid Rotor
Consider two particles of mass m attached to the ends of a massless rigid rod with length a.
The system is free to rotate in 3 dimensions about the center, but the center point is held fixed.
(a) Show that the allowed energies for this rigid rotor are:
n?n(n+1)
En =
where n = 0, 1, 2...
та?
(b) What are the normalized eigenfunctions for this system?
(c) A nitrogen molecule (N2) can be described as a rigid rotor. If the distance between N
atoms is 1.1 Å, then what wavelength of photon must be absorbed to produce a
transition from the n = 1 to n = 4 state?
Starting with the equation of motion of a three-dimensional isotropic harmonic
ocillator
dp.
= -kr,
dt
(i = 1,2,3),
deduce the conservation equation
dA
= 0,
dt
where
1
P.P, +kr,r,.
2m
(Note that we will use the notations r,, r2, r, and a, y, z interchangeably, and similarly
for the components of p.)
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
Chapter 4 Solutions
Introduction To Quantum Mechanics
Ch. 4.1 - Prob. 4.1PCh. 4.1 - Prob. 4.3PCh. 4.1 - Prob. 4.4PCh. 4.1 - Prob. 4.5PCh. 4.1 - Prob. 4.6PCh. 4.1 - Prob. 4.7PCh. 4.1 - Prob. 4.8PCh. 4.1 - Prob. 4.9PCh. 4.1 - Prob. 4.10PCh. 4.1 - Prob. 4.11P
Ch. 4.2 - Prob. 4.12PCh. 4.2 - Prob. 4.13PCh. 4.2 - Prob. 4.14PCh. 4.2 - Prob. 4.15PCh. 4.2 - Prob. 4.16PCh. 4.2 - Prob. 4.17PCh. 4.2 - Prob. 4.18PCh. 4.2 - Prob. 4.19PCh. 4.2 - Prob. 4.20PCh. 4.3 - Prob. 4.21PCh. 4.3 - Prob. 4.22PCh. 4.3 - Prob. 4.23PCh. 4.3 - Prob. 4.24PCh. 4.3 - Prob. 4.25PCh. 4.3 - Prob. 4.26PCh. 4.3 - Prob. 4.27PCh. 4.4 - Prob. 4.28PCh. 4.4 - Prob. 4.29PCh. 4.4 - Prob. 4.30PCh. 4.4 - Prob. 4.31PCh. 4.4 - Prob. 4.32PCh. 4.4 - Prob. 4.33PCh. 4.4 - Prob. 4.34PCh. 4.4 - Prob. 4.35PCh. 4.4 - Prob. 4.36PCh. 4.4 - Prob. 4.37PCh. 4.4 - Prob. 4.38PCh. 4.4 - Prob. 4.39PCh. 4.4 - Prob. 4.40PCh. 4.4 - Prob. 4.41PCh. 4.5 - Prob. 4.42PCh. 4.5 - Prob. 4.43PCh. 4.5 - Prob. 4.44PCh. 4.5 - Prob. 4.45PCh. 4 - Prob. 4.46PCh. 4 - Prob. 4.47PCh. 4 - Prob. 4.48PCh. 4 - Prob. 4.49PCh. 4 - Prob. 4.50PCh. 4 - Prob. 4.51PCh. 4 - Prob. 4.52PCh. 4 - Prob. 4.53PCh. 4 - Prob. 4.54PCh. 4 - Prob. 4.55PCh. 4 - Prob. 4.56PCh. 4 - Prob. 4.57PCh. 4 - Prob. 4.58PCh. 4 - Prob. 4.59PCh. 4 - Prob. 4.61PCh. 4 - Prob. 4.62PCh. 4 - Prob. 4.63PCh. 4 - Prob. 4.64PCh. 4 - Prob. 4.65PCh. 4 - Prob. 4.66PCh. 4 - Prob. 4.70PCh. 4 - Prob. 4.72PCh. 4 - Prob. 4.73PCh. 4 - Prob. 4.75PCh. 4 - Prob. 4.76P
Knowledge Booster
Similar questions
- The angular momentum operator is given by Î = î × p. (a) Assuming we are in cartesian space, prove that [Li, Lj]— = iħ€¡jk (b) Define the ladder operators as L₁ = L₁ iL2. Show that L² = L + L= F ħL3 + L². kLk.arrow_forwardConsider the problem: [cput = (Koux)x+au, 0arrow_forwardLegrende polynomials The amplitude of a stray wave is defined by: SO) =x (21+ 1) exp li8,] sen 8, P(cos 8). INO Here e is the scattering angle, / is the angular momentum and 6, is the phase shift produced by the central potential that performs the scattering. The total cross section is: Show that: 'É4+ 1)sen² 8, .arrow_forwardIn terms of the totally antisymmetric E-symbol (Levi-Civita tensor) with €123 = +1, the vector product can be written as (A x B) i = tijk Aj Bk, where i, j, k = 1, 2, 3 and summation over repeated indices (here j and k) is implied. i) ii) iii) iv) For general vectors A and B, using (2) prove the following relations: a) A x B=-B x A b) (A x B) A = (A x B) - B = 0. The Levi-Civita symbol is related to the Kronecker delta. Prove the following very useful formula €ijk€ilm = 8j18km - Sjm³ki. (2) Prove the formula (3) €imn€jmn = 2dij. Assuming that (3) is true (and using antisymmetry of the E-symbol), prove the relation A x (B x C) = (AC) B- (AB) Carrow_forwardThe Hamilton function for a point particle moving in a central potential is given by p? H + a|x|". 2m Consider the vector A = p x L+ ma where L is the angular momentum of the particle. (a) Calculate the Poisson bracket {H, Ar}, where A is the k-th component of the vector A. (b) Determine the value of the exponent n for which the vector A becomes a conserved quantity.arrow_forwardLet V = 6+ j8 and I = -(2 + j3). (a) Express V and I in phasor form, and find (b) VI, (c) VI*, (d) V/I, and (e) VV/I.Note: express results of (b) - (e) in both complex numbers (with real and imaginary parts) and phasor forms (with amplitude and phase angle). %3Darrow_forward(1) Working with Y" (0, p) show that Y (0, 4) and Y(0,4) functions are are each normalized and that these two orthogonal.arrow_forwardConvert points P(1,3,5), T(0, -4, 3), and S(-3,-4,-10) from Cartesian to cylindrical and spherical coordinates.arrow_forwardConsider 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_forward(d) A linear perturbation A' = nx is applied to the system. What are the first order energy corrections to the energy eigenvalues E? (e) An anharmonic energy perturbation is applied to the system such that H' nx*. What %3D is the first order energy correction E for the ground state |0)? NOTE: Only do the ground state!!!arrow_forward(a) Show, for an appropriate value of a which you should find, that Vo(x) = Ae-αx² is an eigenfunction of the time-independent Schrödinger equation with potential 1 V(x) = -mw²x². Hence find the associated energy eigenvalue. Using the standard integral L -Mr² dx ㅠ M' M > 0, find the value of A. (b) The ladder operators associated with a particle under the potential V(x), defined in part (a), are given by 1 â= = = √ (2 + 1²² ) , &t = √ (2-14). dx dx where L²= h/mw. i. Show that âo(x)=0 where Vo(x) is the function in part (a). What is the physical interpretation of this result? ii. Derive the commutation relations for the ladder operators â and at.arrow_forwardLet V = 2xyz3 +3 In(x² + 2y² + 3z²) V in free space. Evaluate each of the following quantities at P(3, 2, –1): (a) V; (b) |V|; (c) E; (d) |E|; (e) a (f) D.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
Recommended textbooks for you
Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage Learning
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning