Introduction to quantum field theory/Central Potential

Introduction to quantum field theory

Mathematical Prerequisites

• Calculus in curvilinear coordinates, specifically spherical polar coordinates
• Separation of variables in PDE's
• Legendre and Associated Legendre Functions: At the level of Arfken & Weber, including solving the Legendre differential equation by using a series expansion. For a quick refresher, see Lectures by Hamid Meziani (Florida Int. Univ.), specifically Lecture 13 on the topic.

Exercises

Exercise AB1

Atri B. 10:21, 21 September 2017 (UTC)

1. Explicitly transforming from the Cartesian coordinates, show that in the Spherical-Polar coordinate system [math]\displaystyle{ (r, \theta, \varphi) }[/math],
   [math]\displaystyle{ \nabla^2_{(r,\theta,\varphi)} = \frac{1}{r^2}\left( \frac{\partial}{\partial r} r^{2} \frac{\partial}{\partial r} \right) + \frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial \theta} \right) + \frac{1}{r^2\sin^2\! \theta} \frac{\partial^2}{\partial \varphi^2}. }[/math]
2. Show by explicitly constructing a series solution [math]\displaystyle{ P_{\ell}(x) = \sum_{n=0}^{\infty} a_{n}x^{n} }[/math] to the Legendre equation:
   [math]\displaystyle{ \frac{d}{dx}\left[ \left( 1-x^2 \right) \frac{dP_{\ell}(x)}{dx} \right] + \ell \left( \ell+1 \right) P_{\ell}(x) = 0 }[/math]
   that the series terminates if and only if [math]\displaystyle{ \ell \in \mathbb{Z} }[/math]. This is also true for the Associated Legendre functions, [math]\displaystyle{ P_{\ell}^{m}(x) }[/math], which may be obtained by repeatedly differentiating the Legendre polynomials.
3. Show that for a fixed [math]\displaystyle{ \ell }[/math], [math]\displaystyle{ P_{\ell}^{m}(x) }[/math] and [math]\displaystyle{ P_{\ell}^{m^{\prime}}(x) }[/math] are orthogonal to each other for any [math]\displaystyle{ m \neq m^{\prime} }[/math]. The Rodrigues' formula for Associated Legendre functions: [math]\displaystyle{ P_{\ell}^{m}(x) = \frac{(-1)^m}{2^{\ell} \ell!}\left( 1-x^2 \right)^{m/2}\frac{d^{\ell+m}}{dx^{\ell+m}} \left( x^2-1 \right)^\ell }[/math] may be useful.

Exercise AB2

16:37, 12 October 2017 (UTC)

1. • Show by explicit computation in the Cartesian coordinates, that the three components of the angular momentum operator do not commute amongst each other, but instead: [math]\displaystyle{ \left[ \hat{L}_x, \hat{L}_y \right] = i \hbar \hat{L}_z }[/math], and so on.
   • Deduce the uncertainty principle for any two components of the angular momentum, e.g. [math]\displaystyle{ \hat{L}_{x}, \hat{L}_{y} }[/math].
   • In Problem Set 1 you were asked to prove [math]\displaystyle{ e^{\hat{A}} e^{\hat{B}} = e^{\hat{A}+\hat{B}}e^{\frac{1}{2}\left[ \hat{A},\hat{B} \right]} }[/math]. By explicitly computing the LHS and RHS of the equation [math]\displaystyle{ e^{\alpha \hat{A}} e^{\alpha \hat{B}} = e^{\alpha(\hat{A}+\hat{B})}e^{\frac{1}{2}\alpha\left[ \hat{A}, \hat{B} \right]} }[/math] up to the second order in [math]\displaystyle{ \alpha }[/math] (i.e up to [math]\displaystyle{ \alpha^2 }[/math]), show that this is manifestly untrue for the angular momenta operators, e.g. [math]\displaystyle{ \hat{L}_{x} }[/math] and [math]\displaystyle{ \hat{L}_{y} }[/math]. What would be the special condition for two generic operators [math]\displaystyle{ \hat{A} }[/math] and [math]\displaystyle{ \hat{B} }[/math] under which this equality will indeed be true?
2. Show by explicit transformation from the Cartesian coordinates [math]\displaystyle{ (x, y, z) }[/math] (where [math]\displaystyle{ L_z = x \frac{\hbar}{\imath} \frac{\partial}{\partial y} - y \frac{\hbar}{\imath} \frac{\partial}{\partial x} }[/math], and so on) to the Spherical-Polar coordinates [math]\displaystyle{ (r, \theta, \varphi) }[/math], that
   [math]\displaystyle{ \mathbf{L}^2 \equiv L_{x}^2 + L_{y}^2 + L_{z}^2 = -\frac{\hbar^2}{\sin^2\!\theta}\left[ \sin\theta \frac{\partial}{\partial\theta} \left( \sin\theta \frac{\partial}{\partial \theta} \right) + \frac{\partial^2}{\partial\varphi^2} \right]. }[/math]
3. Using the forms of the angular momentum raising and lowering operators [math]\displaystyle{ L_{\pm} }[/math] in the Spherical-Polar coordinates, and the definition of spherical harmonics,
   [math]\displaystyle{ Y_{\ell}^{m}(\theta,\varphi) = (-1)^{m}\sqrt{\frac{2\ell+1}{4\pi}\frac{(l-m)!}{(l+m)!}}P_{\ell}^{m}(\cos\theta)e^{\imath m \varphi}\,, }[/math]
   show that
   [math]\displaystyle{ L_{+} Y_{\ell}^{m}(\theta,\varphi) = \hbar \sqrt{\ell(\ell+1) - m(m+1)}\, Y_{\ell}^{m+1}(\theta,\varphi) }[/math]
   [math]\displaystyle{ L_{-} Y_{\ell}^{m}(\theta,\varphi) = \hbar \sqrt{\ell(\ell+1) - m(m-1)}\, Y_{\ell}^{m-1}(\theta,\varphi). }[/math]
   Recall that [math]\displaystyle{ L_{\pm} = \hbar e^{\pm \imath \varphi}\left[ \pm\frac{\partial}{\partial \theta} + \imath \cot\theta \frac{\partial}{\partial \varphi} \right]. }[/math]