Category Archives: Mathematical Physics

Self-Adjoint Differential Equations III: Real Eigenvalues, Gram-Schmidt Orthogonalization

In here, I mentioned that the eigenvalues of a Hermitian operator are real and that the eigenfunctions of a Hermitian operator are orthogonal. Let $\mathcal{L}$ be a Hermitian operator and let $u_i$, $u_j$ be eigenfunctions of $\mathcal{L}$ with eigenvalues $\lambda_i$, … Continue reading

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Self-Adjoint Differential Equations II: Hermitian Operators

Let $\mathcal{L}$ be a second-order self-adjoint differential operator. Then $\mathcal{L}u(x)$ may be written as $$\mathcal{L}u(x)=\frac{d}{dx}\left[p(x)\frac{du(x)}{dx}\right]+q(x)u(x)\ \ \ \ \ (1)$$ as we discussed here. Multiply (1) by $v^\ast$ ($v^\ast$ is the complex conjugate of $v$) and integrate \begin{align*} \int_a^bv^\ast\mathcal{L}udx&=\int_a^bv^\ast\frac{d}{dx}\left[p(x)\frac{du(x)}{dx}\right]dx+\int_a^bv^\ast qudx\\ … Continue reading

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Self-Adjoint Differential Equations I

Let $\mathcal{L}$ be the second-order linear differential operator $$\mathcal{L}=p_0(x)\frac{d^2}{dx^2}+p_1(x)\frac{d}{dx}+p_2(x)$$ which acts on a function $u(x)$ as $$\mathcal{L}u(x)=p_0(x)\frac{d^2u(x)}{dx^2}+p_1(x)\frac{du(x)}{dx}+p_2(x)u(x).\ \ \ \ \ (1)$$ Define an adjoint operator $\bar{\mathcal{L}}$ by \begin{align*} \bar{\mathcal{L}}&:=\frac{d^2}{dx^2}[p_0u]-\frac{d}{dx}[p_1u]+p_2u\\ &=p_0\frac{d^2u}{dx^2}+(2p_0^\prime-p_1)\frac{du}{dx}+(p_0^{\prime\prime}-p_1+p_2)u. \end{align*} If $\mathcal{L}=\bar{\mathcal{L}}$, $\mathcal{L}$ is said to be self-adjoint. … Continue reading

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Legendre Functions III: Special Values, Parity, Orthogonality

Special Values From the generating function $$g(x,t)=\frac{1}{(1-2xt+t^2)^{1/2}},$$ when $x=1$ we obtain \begin{align*} g(1,t)&=\frac{1}{(1-2t+t^2)^{1/2}}\\ &=\frac{1}{1-t}\\ &=\sum_{n=0}^\infty t^n, \end{align*} since $|t|<1$. On the other hand, $$g(1,t)=\sum_{n=0}^\infty P_n(1)t^n.$$ So by comparison we get $$P_n(1)=1.$$ Similarly, if we let $x=-1$, $$P_n(-1)=(-1)^n.$$ For $x=0$, the … Continue reading

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Legendre Functions II: Recurrence Relations and Special Properties

In this lecture, we derive some important recurrence relations of Legendre functions and use them to show that Legendre functions are indeed solutions of a differential equation, called Legendre’s differential equation. Differentiating the generating function $$g(x,t)=(1-2xt+t^2)^{-1/2}=\sum_{n=0}^\infty P_n(x)t^n,\ |t|<1\ \ \ … Continue reading

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Tensors I

Tensors may be considered as a generalization of vectors and covectors. They are extremely important quantities for studying differential geometry and physics. Let $M^n$ be an $n$-dimensional differentiable manifold. For each $x\in M^n$, let $E_x=T_xM^n$, i.e. the tangent space to … Continue reading

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Helmholtz Equation

Helmholtz equation $$\nabla^2\psi+k^2\psi=0\ \ \ \ \ \mbox{(1)}$$ is extremely important in physics. Solving many physically important partial differential equations such as heat equation, wave equation (Klein-Gordon equation), Maxwell’s equations, and Schr√∂dinger equation, etc. often require solving Helmholtz equation (1). … Continue reading

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Legendre Functions I: A Physical Origin of Legendre Functions

Consider an electric charge $q$ placed on the $z$-axis at $z=a$. The electrostatic potential of charge $q$ is $$\varphi=\frac{1}{4\pi\epsilon_0}\frac{q}{r_1}.\ \ \ \ \ \mbox{(1)}$$ Using the Laws of Cosine, one can write $r_1$ in terms of $r$ and $\theta$: $$r_1=\sqrt{r^2+a^2-2ar\cos\theta}$$ … Continue reading

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Spherical Bessel Functions

When the Helmholtz equation is separated in spherical coordinates the radial equation has the form $$r^2\frac{d^2R}{dr^2}+2r\frac{dR}{dr}+[k^2r^2-n(n+1)]R=0.\ \ \ \ \ \mbox{(1)}$$ The equation (1) looks similar to Bessel’s equation.¬† If we use the transformation $R(kr)=\frac{Z(kr)}{(kr)^{1/2}}$, (1) turns into Bessel’s equation … Continue reading

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Neumann Functions, Bessel Function of the Second Kind $N_\nu(X)$

In here, we have considered Bessel functions $J_\nu(x)$ for $\nu=\mbox{integer}$ case only. Note that $J_\nu$ and $J_{-\nu}$ are linearly independent if $\nu$ is a non-integer. If $\nu$ is an integer, $J_\nu$ and $J_{-\nu}$ satisfy the relation $J_{-\nu}=(-1)^\nu J_\nu$, i.e. they … Continue reading

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