2025-02-18
Solutions to Principles of Quantum Mechanics
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2024-08-17
Solutions to Principles of Quantum Mechanics
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Chapter 5 Simple Problems in One Dimension

5.1 The Free Particle

Exercise 5.1.1 Show that Eq. (5.1.9) may be rewritten as an integral over EE and a sum over the ±\pm index as

U(t)=α=±0[m(2mE)1/2]E,αE,αeiEt/dEU(t)=\sum_{\alpha= \pm} \int_0^{\infty}\left[\frac{m}{(2 m E)^{1 / 2}}\right]|E, \alpha\rangle\langle E, \alpha| \mathrm{e}^{-\mathrm{i} E t / \hbar} \mathrm{d} E
2024-08-05
Solutions to Principles of Quantum Mechanics
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Chapter 4 The Postulates——a General Discussion

4.1 The Postulates

4.2 Discussion of Postulates I-III

Exercise 4.2.1 Consider the following operators on a Hilbert space V3(C)\mathbb{V}^{3}(C):

Lx=121/2(010101010)Ly=121/2(0i0i0i0i0)Lz=(100000001)L_{x}=\frac{1}{2^{1/2}}\begin{pmatrix} 0&1&0\\ 1&0&1\\ 0&1&0 \end{pmatrix}\qquad L_{y}=\frac{1}{2^{1/2}}\begin{pmatrix} 0&-i&0\\ i&0&-i\\ 0&i&0 \end{pmatrix}\qquad L_{z}=\begin{pmatrix} 1&0&0\\ 0&0&0\\ 0&0&-1 \end{pmatrix}

(1) What are the possible values one can obtain if LzL_{z} is measured?

(2) Take the state in which Lz=1L_{z}=1. In this state what are Lx\langle L_{x}\rangle, Lx2\langle L_{x}^{2}\rangle and ΔLx\Delta L_{x}?

(3) Find the normalized eigenstates and the eigenvalues of LxL_{x} in the LzL_{z} basis.

(4) If the particle is in the state with Lz=1L_{z}=-1, and LxL_{x} is measured, what are the possible outcomes and their probabilities? (5) Consider the state

ψ=(1/21/21/21/2)|\psi\rangle=\begin{pmatrix} 1/2\\ 1/2\\ 1/2^{1/2} \end{pmatrix}

in the LzL_{z} basis. If Lz2L_{z}^{2} is measured in this state and a result +1+1 is obtained, what is the state after the measurement? How probable was this result? If LzL_{z} is measured, what are the outcomes and respective probabilities?

(6) A particle is in a state for which the probabilities are P(Lz=1)=1/4P(L_{z}=1)=1/4, P(Lz=0)=1/2P(L_{z}=0)=1/2, and P(Lz=1)=1/4P(L_{z}=-1)=1/4. Convince yourself that the most general, normalized state with this property is

ψ=eiδ12Lz=1+eiδ221/2Lz=0+eiδ32Lz=1|\psi\rangle=\frac{e^{i \delta_1}}{2}\left|L_z=1\right\rangle+\frac{e^{i \delta_2}}{2^{1 / 2}}\left|L_z=0\right\rangle+\frac{e^{i \delta_3}}{2}\left|L_z=-1\right\rangle

It was stated earlier on that if ψ|\psi\rangle is a normalized state then the state eiθψe^{i \theta}|\psi\rangle is a physically equivalent normalized state. Does this mean that the factors eiδie^{i \delta_i} multiplying the LzL_z eigenstates are irrelevant? [Calculate for example P(Lx=0)P\left(L_x=0\right).]

2024-07-31
Solutions to Principles of Quantum Mechanics
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Chapter 3 All Is Not Well with Classical Mechanics

3.1 Particles and Waves in Classical Physics

2024-07-11
Solutions to Principles of Quantum Mechanics
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Chapter 2 Review of Classical Mechanics

2.1 The Principle of Least Action and Lagrangian Mechanics

Exercise 2.1.1 Consider the following system, called a harmonic oscillator. The block has a mass mm and lies on a frictionless surface. The spring has a force constant kk. Write the Lagrangian and get the equation of motion.