Mathematical Physics II
A continuous, derivation-rich journey through the core mathematical methods of modern physics. Emphasis is placed on the method of Frobenius and the classical special functions (Legendre, Hermite, Laguerre, Bessel).
Learning Philosophy
Theory is interleaved with full derivations, worked examples, and typical examination-style problems. The Frobenius method is developed carefully so that the classical orthogonal polynomials and Bessel functions appear as natural consequences of regular singular points.
Syllabus Roadmap
Fourier Series
Periodic functions, orthogonality, Dirichlet conditions, complex form, Parseval, applications.
Frobenius & Special Functions
Full derivations of Legendre, Hermite, Laguerre, Bessel · Rodrigues, generating functions, orthogonality.
Dirac Delta Function
Definition, representations, properties, distributions.
Special Integrals
Beta & Gamma functions, Error function, Stirling.
Theory of Errors
Systematic vs random, propagation, normal law.
Partial Differential Equations
Laplace & Wave equations · separation of variables · special-function solutions.
Fourier Series
1.1 Periodic Functions & Orthogonality
A function \( f(x) \) is periodic with period \( T \) if \( f(x+T) = f(x) \) for all \( x \) in the domain. The fundamental period is the smallest positive such \( T \). The trigonometric system \(\{1,\cos nx,\sin nx\}\) is orthogonal on \([-\pi,\pi]\).
Orthogonality Relations
1.2 Dirichlet Conditions
If \( f \) is \( 2\pi \)-periodic, absolutely integrable on one period, and possesses only finitely many maxima, minima and jump discontinuities, then the Fourier series converges to \( f(x) \) at every point of continuity and to the average of the left- and right-hand limits at a jump discontinuity.
1.3 Real Fourier Series
1.4 Complex Form
The relations \( c_0=a_0/2 \), \( c_n=(a_n-ib_n)/2 \), \( c_{-n}=(a_n+ib_n)/2 \) connect the two forms.
1.5 Even & Odd Extensions
Even: \( f(-x)=f(x) \)
Only cosine terms remain (\( b_n=0 \)). On \([0,L]\) one obtains a Fourier cosine series.
Odd: \( f(-x)=-f(x) \)
Only sine terms remain (\( a_n=0 \)). On \([0,L]\) one obtains a Fourier sine series.
1.6 Parseval’s Identity
A powerful tool for summing numerical series (e.g. \( \sum 1/n^2=\pi^2/6 \)).
1.7 Typical Worked Problems
Example 1 – Square wave
Let \( f(x)=1 \) for \( 0
Example 2 – \( f(x)=x \) on \( (-\pi,\pi) \)
Odd ⇒ only sine terms. Integration by parts yields \( b_n=2(-1)^{n+1}/n \). Thus \[ x=2\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\sin nx. \] Parseval immediately gives \( \sum 1/n^2=\pi^2/6 \).
Typical Problems
- Expand \( f(x)=|x| \) on \( (-\pi,\pi) \) in a Fourier series and deduce \( \sum_{n=0}^\infty 1/(2n+1)^2=\pi^2/8 \).
- Find the Fourier cosine series of \( f(x)=x \) on \( [0,\pi] \) and use Parseval to evaluate \( \sum 1/n^4 \).
- Show that the complex coefficients of \( e^{ax} \) (\( a \) real) on \( (-\pi,\pi) \) are \( c_n=\frac{(-1)^n\sinh(a\pi)}{\pi(a-in)} \).
- A function equals \( x(\pi-x) \) on \( (0,\pi) \). Write its sine series and its cosine series; compare the rates of convergence.
Frobenius Method & Special Functions
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2.1 Classification of Singular Points
Consider the second-order linear equation written in standard form \[ y''+P(x)y'+Q(x)y=0. \] A point \( x_0 \) is an ordinary point if both \( P \) and \( Q \) are analytic there. Otherwise it is a singular point. The singularity is regular if \( (x-x_0)P(x) \) and \( (x-x_0)^2 Q(x) \) are analytic; otherwise it is irregular.
Near a regular singular point the method of Frobenius guarantees at least one solution of the form \( x^r\sum a_n x^n \) (with \( a_0\neq0 \)). The possible values of the indicial exponent \( r \) are roots of a quadratic equation.
2.2 The Method of Frobenius – General Procedure
Assume a regular singular point at the origin. Substitute the ansatz \[ y=x^r\sum_{n=0}^\infty a_n x^n,\qquad a_0\neq0 \] into the differential equation. Collect the lowest power of \( x \) to obtain the indicial equation \[ r(r-1)+p_0 r+q_0=0, \] where \( p_0=\lim_{x\to0}xP(x) \) and \( q_0=\lim_{x\to0}x^2 Q(x) \). The two roots \( r_1\geq r_2 \) determine the possible leading behaviours. A recurrence relation then fixes all higher coefficients \( a_n \) in terms of \( a_0 \).
Possible behaviours when \( r_1-r_2 \) is an integer
- If the difference is not an integer, two independent Frobenius series exist.
- If the difference is a positive integer, the larger root always yields a solution; the smaller root may or may not (logarithmic term may appear).
- If the roots are equal, the second independent solution necessarily contains a logarithm.
2.3 Legendre’s Equation – Full Derivation
Legendre’s equation arises in the separation of variables for Laplace’s equation in spherical coordinates: \[ (1-x^2)y''-2xy'+\ell(\ell+1)y=0,\qquad x=\cos\theta\in[-1,1]. \] The points \( x=\pm1 \) are regular singular points; the origin is an ordinary point. We therefore seek a power-series solution about \( x=0 \): \[ y=\sum_{n=0}^\infty a_n x^n. \] Differentiation and substitution produce the two-term recurrence \[ a_{n+2}=\frac{(n+\ell+1)(n-\ell)}{(n+1)(n+2)}a_n. \] Consequently the series splits into an even part (starting from \( a_0 \)) and an odd part (starting from \( a_1 \)). For the series to terminate and remain finite at \( x=\pm1 \), one of the two series must terminate, which forces \( \ell \) to be a non-negative integer. The resulting polynomials, normalised by \( P_\ell(1)=1 \), are the Legendre polynomials.
Rodrigues formula
Proof follows by verifying that the right-hand side satisfies Legendre’s equation and the normalisation \( P_\ell(1)=1 \).
Generating function
Orthogonality
Worked example – first few polynomials via recurrence
Start with \( a_0=1 \), \( \ell=2 \). The recurrence gives \( a_2=-3/2\cdot a_0=-3/2 \), \( a_4=0 \). Normalising so that \( P_2(1)=1 \) yields \( P_2(x)=\frac12(3x^2-1) \).
2.4 Hermite’s Equation – Full Derivation
Hermite’s equation appears in the quantum harmonic oscillator: \[ y''-2xy'+2\nu y=0. \] The origin is an ordinary point, so we may use an ordinary power series \[ y=\sum_{n=0}^\infty a_n x^n. \] Substitution produces the recurrence \[ a_{n+2}=\frac{2(n-\nu)}{(n+1)(n+2)}a_n. \] Termination occurs when \( \nu=N \) (a non-negative integer). The resulting polynomials are the Hermite polynomials \( H_N(x) \), conventionally normalised so that the leading coefficient is \( 2^N \).
Rodrigues formula
Generating function
Orthogonality (with weight \( e^{-x^2} \))
2.5 Laguerre’s Equation – Full Derivation
Laguerre’s equation arises in the radial hydrogen atom problem: \[ xy''+(1-x)y'+n y=0. \] The origin is a regular singular point. The Frobenius ansatz \( y=x^r\sum a_k x^k \) yields the indicial equation \( r^2=0 \), so a double root \( r=0 \). The recurrence becomes \[ a_{k+1}=\frac{k-n}{(k+1)^2}a_k. \] When \( n \) is a non-negative integer the series terminates and we obtain the Laguerre polynomials \( L_n(x) \).
Rodrigues formula
Generating function
Orthogonality (with weight \( e^{-x} \))
2.6 Bessel’s Equation – Full Derivation
Bessel’s equation of order \( \nu \) is \[ x^2 y''+xy'+(x^2-\nu^2)y=0. \] The origin is a regular singular point. The Frobenius ansatz \( y=x^r\sum_{k=0}^\infty a_k x^k \) produces the indicial equation \[ r^2-\nu^2=0\qquad\Rightarrow\qquad r=\pm\nu. \] For the larger root \( r=\nu \) the recurrence is \[ a_{k}=\frac{-1}{k(k+2\nu)}a_{k-2}\qquad(k\geq2). \] Choosing \( a_0=\frac1{2^\nu\Gamma(\nu+1)} \) yields the standard Bessel function of the first kind \[ J_\nu(x)=\sum_{k=0}^\infty\frac{(-1)^k}{k!\,\Gamma(k+\nu+1)}\Bigl(\frac x2\Bigr)^{2k+\nu}. \] When \( \nu \) is not an integer the second independent solution is \( J_{-\nu}(x) \). When \( \nu=n \) is an integer one must introduce the Bessel function of the second kind \( Y_n(x) \) (which contains a logarithmic singularity).
Important recurrence relations
Generating function (integer order)
Worked example – indicial roots for a general equation
For \( 2x^2 y''+x(2x+1)y'-y=0 \) one finds \( p_0=1/2 \), \( q_0=-1/2 \). The indicial equation is \( 2r(r-1)+r-1=0 \), i.e. \( (2r+1)(r-1)=0 \), so \( r=1 \) and \( r=-1/2 \).
Typical Problems – Frobenius & Special Functions
- Apply the method of Frobenius to \( x y''+(1-x)y'+2y=0 \) and obtain the first three non-zero terms of each independent solution.
- Derive the recurrence relation for Legendre polynomials from the generating function and verify \( P_3(x) \).
- Using Rodrigues’ formula, prove the orthogonality of Hermite polynomials with weight \( e^{-x^2} \).
- Show that the series solution of Bessel’s equation of order \( 1/2 \) can be expressed in elementary functions: \[ J_{1/2}(x)=\sqrt{\frac2{\pi x}}\sin x,\qquad J_{-1/2}(x)=\sqrt{\frac2{\pi x}}\cos x. \]
- Find the first four Laguerre polynomials by successive differentiation of the Rodrigues formula and verify that they satisfy Laguerre’s equation.
- Prove the recurrence \( (n+1)L_{n+1}(x)=(2n+1-x)L_n(x)-n L_{n-1}(x) \).
- Locate the regular singular points of the hypergeometric equation and write down its indicial exponents at \( x=0 \).
Dirac Delta Function
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3.1 Definition as a Distribution
The Dirac delta is not a function in the classical sense; it is a continuous linear functional on a suitable space of test functions: \[ \langle\delta_a,f\rangle=\int_{-\infty}^\infty f(x)\delta(x-a)\,dx=f(a). \]
3.2 Common Representations
Gaussian
Rectangular / Lorentzian
3.3 Fundamental Properties
- \( \delta(-x)=\delta(x) \) (even)
- \( x\delta(x)=0 \)
- \( \delta(ax)=\frac1{|a|}\delta(x) \)
- \( \delta(g(x))=\sum_i\frac{\delta(x-x_i)}{|g'(x_i)|} \) where \( x_i \) are simple zeros of \( g \)
- \( \int f(x)\delta'(x)\,dx=-f'(0) \)
- \( \theta'(x)=\delta(x) \) (Heaviside step function)
Example – composition
\( \delta(x^2-a^2)=\frac1{2|a|}[\delta(x-a)+\delta(x+a)] \) for \( a\neq0 \).
Typical Problems
- Evaluate \( \int_{-\infty}^\infty (x^3+2x+1)\delta(x-1)\,dx \) and \( \int_{-\infty}^\infty e^{-x^2}\delta'(x)\,dx \).
- Prove that \( \delta(ax+b)=\frac1{|a|}\delta\bigl(x+b/a\bigr) \).
- Show that the Fourier transform of 1 is \( 2\pi\delta(k) \).
- Express the charge density of a unit point charge at the origin in spherical coordinates using \( \delta(r) \).
Some Special Integrals
4.1 The Gamma Function
Analytic continuation yields a meromorphic function with simple poles at the non-positive integers. The reflection formula reads \[ \Gamma(z)\Gamma(1-z)=\frac\pi{\sin(\pi z)}. \] Stirling’s approximation for large \( |z| \) in \( |\arg z|<\pi-\delta \) is \[ \Gamma(z)\sim\sqrt{2\pi/z}\,(z/e)^z. \]
4.2 The Beta Function
The substitution \( t=\sin^2\theta \) converts many definite integrals into Beta functions.
4.3 The Error Function
Asymptotic expansion for large positive \( x \): \[ \operatorname{erfc}(x)\sim\frac{e^{-x^2}}{x\sqrt\pi}\Bigl(1-\frac1{2x^2}+\frac{1\cdot3}{(2x^2)^2}-\cdots\Bigr). \]
Example – Gaussian integral
\( \int_{-\infty}^\infty e^{-ax^2+bx}\,dx=\sqrt{\pi/a}\,e^{b^2/(4a)} \) (\( a>0 \)). This follows by completing the square and using \( \Gamma(1/2)=\sqrt\pi \).
Typical Problems
- Evaluate \( \Gamma(1/2) \), \( \Gamma(3/2) \) and \( B(3,2) \) from first principles.
- Show that \( \int_0^{\pi/2}\sin^m\theta\cos^n\theta\,d\theta=\frac12 B\bigl(\frac{m+1}2,\frac{n+1}2\bigr) \).
- Prove the duplication formula \( \Gamma(z)\Gamma(z+1/2)=2^{1-2z}\sqrt\pi\,\Gamma(2z) \).
- Obtain the leading asymptotic term of \( \operatorname{erfc}(x) \) for large positive \( x \) by integration by parts.
Theory of Errors
5.1 Systematic versus Random Errors
Systematic
Reproducible biases arising from imperfect calibration, environmental drift, etc. They cannot be reduced by averaging; they must be estimated and corrected.
Random
Unpredictable fluctuations. Their effect is reduced by repeated measurement; the standard error of the mean falls as \( 1/\sqrt N \).
5.2 Propagation of Independent Uncertainties
If \( q=f(x,y,\dots) \) and the measured quantities are independent, the variance of \( q \) is approximately \[ \sigma_q^2\approx\Bigl(\frac{\partial f}{\partial x}\Bigr)^2\sigma_x^2+\Bigl(\frac{\partial f}{\partial y}\Bigr)^2\sigma_y^2+\cdots \] evaluated at the mean values.
5.3 The Normal (Gaussian) Law
The probable error (half-width of the central 50 % probability) is approximately \( 0.6745\,\sigma \). The standard error of the mean of \( N \) independent measurements is \( \sigma/\sqrt N \).
Typical Problems
- A length is measured as \( L=12.34\pm0.05 \) cm and a time as \( T=3.2\pm0.1 \) s. Estimate the uncertainty in the velocity \( v=L/T \).
- Show that for a product \( q=xy \) one has \( (\sigma_q/q)^2=(\sigma_x/x)^2+(\sigma_y/y)^2 \) (relative uncertainties add in quadrature).
- Ten measurements of a quantity yield a sample standard deviation \( s=0.12 \). What is the standard error of the mean? How many measurements are needed to reduce it by a factor of three?
Partial Differential Equations
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6.1 Separation of Variables – General Strategy
Assume a product solution \( u=X(x)Y(y)\dots T(t) \). The PDE splits into ordinary differential equations linked by separation constants. Boundary conditions quantize the allowed values of those constants (eigenvalue problems). The general solution is a linear combination (often an infinite series) of the separated solutions.
6.2 Laplace’s Equation \( \nabla^2 u=0 \)
Cartesian: product of sines/cosines or exponentials; double Fourier series on a rectangle.
Cylindrical: radial equation becomes Bessel’s equation; angular part is trigonometric; axial part exponential or hyperbolic.
Spherical: angular part yields spherical harmonics \( Y_{\ell m}(\theta,\phi) \) (associated Legendre functions); radial part is \( r^\ell \) and \( r^{-(\ell+1)} \).
Example – Dirichlet problem on a disk
Inside the unit disk with boundary data \( u(1,\theta)=f(\theta) \), the solution is the Poisson integral \[ u(r,\theta)=\frac{1-r^2}{2\pi}\int_0^{2\pi}\frac{f(\phi)}{1-2r\cos(\theta-\phi)+r^2}\,d\phi, \] which is equivalent to the Fourier series \( \sum(A_n r^n\cos n\theta+B_n r^n\sin n\theta) \).
6.3 The Wave Equation \( \partial_t^2 u=c^2\nabla^2 u \)
Finite string with fixed ends → ordinary Fourier sine series in space, harmonic time dependence. Rectangular membrane → double Fourier series. Circular membrane → Bessel functions \( J_m(k_{mn}r) \) whose zeros determine the eigenfrequencies.
Typical Problems
- Solve Laplace’s equation in a unit square with three sides held at zero and the fourth side equal to \( \sin(\pi x) \).
- Find the frequencies of a circular membrane of radius \( a \) for the modes \( (0,1) \), \( (1,1) \) and \( (2,1) \). (The first zeros of \( J_0,J_1,J_2 \) are approximately 2.405, 3.832, 5.136.)
- Separate variables in spherical coordinates for the Helmholtz equation and show that the radial equation becomes the spherical Bessel equation.
- A string of length \( \pi \) is released from rest with initial displacement \( x(\pi-x) \). Write the solution of the wave equation as a Fourier sine series.
🏆 Comprehensive Problems & Challenges
A selection of examination-style problems that integrate the material of the whole course.
Mixed Problems
- Expand \( f(x)=x^2 \) on \( (-\pi,\pi) \) in a Fourier series. Use Parseval’s identity to evaluate \( \sum_{n=1}^\infty 1/n^4 \).
- Starting from the generating function of the Legendre polynomials, derive the recurrence \( ( \ell+1)P_{\ell+1}=(2\ell+1)x P_\ell-\ell P_{\ell-1} \).
- Show that the Hermite polynomials satisfy the differential equation obtained by applying the operator \( D^2-2xD \) to the generating function.
- Prove that \( J_{1/2} \) and \( J_{-1/2} \) form a basis of solutions of Bessel’s equation of order \( 1/2 \), and express them in elementary functions.
- Using the Rodrigues formula for Laguerre polynomials, evaluate \( \int_0^\infty x L_n(x)L_m(x)e^{-x}\,dx \).
- A circular drum of radius 1 is struck so that the initial velocity is zero and the initial displacement is \( (1-r^2) \). Expand the subsequent motion in Bessel modes (axisymmetric case).
- Evaluate \( \int_0^\infty x^{a-1}e^{-px}\cos qx\,dx \) in terms of the Gamma function (assume \( \Re a>0 \), \( \Re p>|q| \)).