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 $y$.

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 $E$ and a sum over the $\pm$ index as

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 $\mathbb{V}^{3}(C)$:

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

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

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

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

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

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

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