Chapter 10 Systems with $N$ Degrees of Freedom

10.1 $N$ Particles in One Dimension

Exercise 10.1.1 Show the following:

(1)

(2)

(3)If

then

and similarly with $1\to 2$.

(4)

Chapter 9 The Heisenberg Uncertainty Relations

9.1 Introduction

9.2 Derivation of the Uncertainty Relations

9.3 The Minimum Uncertainty Packet

9.4 Applications of the Uncertainty Principle

Exercise 9.4.1 Consider the oscillator in the state $|n=1\rangle$ and verify that

Chapter 8 The Path Integral Formulation of Quantum Theory

8.1 The Path Integral Recipe

8.2 Analysis of the Recipe

8.3 An Approximation to $U(t)$ for the Free Particle

8.4 Path Integral Evaluation of the Free-Particle Propagator

8.5 Equivalence to the Schrödinger Equation

8.6 Potentials of the Form $V=a+bx+cx^{2}+d\dot{x}+ex\dot{x}$

Exercise 8.6.1 Verify that

agrees with the exact result, Eq. (5.4.31), for $V(x)=-fx$. Hint: Start with $x_{\text{cl}}(t^{\prime\prime})=x_{0}+v_{0}t^{\prime\prime}+\dfrac{1}{2}(f/m)t^{\prime\prime 2}$ and find the constants $x_{0}$ and $v_{0}$ from the requirement that $x_{\text{cl}}(0)=x^{\prime}$ and $x_{\text{cl}}(t)=x$.

Chapter 7 The Harmonic Oscillator

7.1 Why Study the Harmonic Oscillator?

7.2 Review of the Classical Oscillator

7.3 Quantization of the Oscillator (Coordinate Basis)

Exercise 7.3.1 Consider the question why we tried a power-series solution for Eq. (7.3.11) but not Eq. (7.3.8). By feeding in a series into the latter, verify that a three-term recursion relation between $C_{n+2}$, $C_{n}$, and $C_{n-2}$ obtains, from which the solution does not follow so readily. The problem is that $\psi^{\prime\prime}$ has two powers of $y$ less than $2 \varepsilon \psi$, while the $-y^{2}$ piece has two more powers of $y$. In Eq. (7.3.11) on the other hand, of the three pieces $u^{\prime\prime}$, $-2yu^{\prime}$, and $(2\varepsilon-1)u$, the last two have the same powers of $y$.