Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 10.4, Problem 10.9P
To determine
Whether the ground state of hydrogen satisfies the integral form of the Schrödinger equation, for the appropriate
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Consider the function
v(1,2) =(
[1s(1) 3s(2) + 3s(1) 1s(2)]
[x(1) B(2) + B(1) a(2)]
Which of the following statements is incorrect concerning p(1,2) ?
a.
W(1,2) is normalized.
Ob.
The function W(1,2) is symmetric with respect to the exchange of the space and the spin coordinates of the two electrons.
OC.
y(1,2) is an eigenfunction of the reference (or zero-order) Hamiltonian (in which the electron-electron repulsion term is ignored) of Li with
eigenvalue = -5 hartree.
d.
The function y(1,2) is an acceptable wave function to describe the properties of one of the excited states of Lit.
Oe.
The function 4(1,2) is an eigenfunction of the operator S,(1,2) = S;(1) + S,(2) with eigenvalue zero.
Demonstrate that e#ikz are solutions to both H and p, (momentum) for a free particle. Do you expect a
difference for a bound particle where V (z) + 0?
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.)
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- Solve the “particle in a box” problem to find Ψ(x, t) if Ψ(x, 0) = 1 on (0, π). What is En? The function of interest here which you should plot is |Ψ(x, t)|2.arrow_forwardThe harmonic oscillator eigenfunction ₁(x) = = 4a³\1/4 e-ax²/2, where a = μω ħ (a) Find (x²) for an oscillator in this state, and express your result in terms of u, w, and ħ. 1 (b) In an eigenstate it is always true that (T) = (V) for a harmonic oscillator. If ↑ =-=-=-=A² 2μ 1 and ✩ = µw²x², find (p²) when the system is in the state 1₁ (x). (Note: You can use the result found in part (a), or compute it directly from the form of the operator p.) 2arrow_forwardConsider a composite state of an electron with total angular momentum j1 = 1/2 and a proton with angular momentum j2 = 3/2. Find all the eigenstates of |j1,j2;j,m⟩ as the linear combination of product states of angular momentum of electron and proton |j1,j2;m1,m2⟩. Give the values of Clebsch-Gordon coefficients you get from here. If the system is found in state |j1 = 1/2,j2 = 3/2;j = 1,m = −1⟩, what is the probability that j1z = −1/2 and what is the probability that j1z = 1/2arrow_forward
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- Consider the Schrodinger equation for a one-dimensional linear harmonic oscillator: -(hbar2/2m) * d2ψ/dx2 + (kx2/2)*ψ(x) = Eψ(x) Substitute the wavefunction ψ(x) = e-(x^2)/(ξ^2) and find ξ and E required to satisfy the Schrodinger equation. [Hint: First calculate the second derivative of ψ(x), then substitute ψ(x) and ψ′′(x). After this substitution, there will be an overall factor of e-(x^2)/(ξ^2) on both sides of the equation which canbe an canceled out. Then, gather all terms which depend on x into one coefficient multiplying x2. This coefficient must be zero because the equation must be satisfied for any x, and equating it with zero yields the expression for ξ. Finally, the remaining x-independent part of the equation determines the eigenvalue for energy E associated with this solution.]arrow_forwardThe following problem arises in quantum mechanics (see Chapter 13, Problem 7.21). Find the number of ordered triples of nonnegative integers a, b, c whose sum a+b+c is a given positive integer n. (For example, if n = 2, we could have (a, b, c) = (2, 0, 0) or (0, 2, 0) or (0, 0, 2) or (0, 1, 1) or (1, 0, 1) or (1, 1, 0).) Hint: Show that this is the same as the number of distinguishable distributions of n identical balls in 3 boxes, and follow the method of the diagram in Example 5.arrow_forwardFor a particle in a 1-dimensional infinitely deep box of length L, the normalized wave function or the 1st excited state can be written as: Ψ2(x) = {1/i(2L)1/2} ( eibx -e-ibx), where b = 2π/L. Give the full expression that you need to solve to determine the probalibity of finding the particle in the 1st third of the box. Simplify as much as possible but do not solve any integrals.arrow_forward
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