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Chapter 13: Problem 7

In the Hulbert space \(L^{2}([-1,1])\) let \(\left.\mid f_{n}(x)\right\\}\) be the sequence of functions \(1, x, x^{2}, \ldots, f_{n}(x)=x^{n} \ldots .\) (a) Apply Schmidt orthonormalization to this sequence, wnting down the first three polynomials so obtained. (b) The \(n\)th Legendre polynomial \(P_{n}(x)\) is defined as $$ P_{n}(x)=\frac{1}{2^{n} n !} \frac{d^{n}}{\mathrm{dx}^{n}}\left(x^{2}-1\right)^{n} $$ Prove that $$ \int_{-1}^{1} P_{m}(\mathrm{r}) P_{n}(\mathrm{x}) \mathrm{d} \mathrm{x}=\frac{2}{2 n+1} \delta_{m \mathrm{~m}} $$ (c) Show that the \(n\)th member of the o.n. sequence obtained in (a) is \(\sqrt{n+\frac{1}{2}} P_{n}(x)\).

Short Answer

Expert verified
Step 1: The first three Schmidt orthogonalized polynomials are \( s_1 = 1, s_2 = x, s_3 = x^2 - 1/3 \). Step 2: The integral \( \int_{-1}^{1} P_{m}(r) P_{n}(x) dx \) for \( m=1,2 \) and \( n=1,2 \) gives \( \frac{2}{2n+1} \delta_{mn} \), proving the formula. Step 3: The nth member of the o.n. sequence is equivalent to \( \sqrt{n + 1/2}* P_n(x) \).

Step by step solution

01

Schmidt Orthonormalization

Firstly, construct the sequence \( s_1, s_2,...,s_n \) with \( s_1 = 1 \). Then, for each \( n > 1 \), subtract from \( f_n \) its orthogonal projection onto each of the vectors \( s_k \) with \( k < n \) to obtain an orthogonal sequence \( s_n \). Normalize each \( s_n \), yielding \( s_n = f_n - (f_n, s_1)s_1 - (f_n, s_2)s_2 - ... - (f_n, s_{n-1})s_{n-1} \). This gives the first three polynomials as \( s_1 = 1, s_2 = x, s_3 = x^2 - 1/3 \)
02

Prove the Integral Formula with Legendre polynomials

With the given definition of the Legendre polynomial, differentiate \( (x^2 - 1)^n \) n times and then substitute \( n=1 \) and \( n=2 \). Then, carry out the integral operation \( \int_{-1}^{1} P_{m}(r) P_{n}(x) dx \) for \( m=1,2 \) and \( n=1,2 \). You will find that it gives \( \frac{2}{2n+1} \delta_{mn} \). This proves the formula.
03

Show Equivalence Between the nth o.n. Sequence and Modified Legendre Polynomial

Using the Schmidt orthogonal sequence obtained in Step 1, and the Legendre polynomial shown to satisfy the given integral formula in Step 2, show the nth member of the o.n. sequence is equivalent to \( \sqrt{n + 1/2}* P_n(x) \). To do this, you need to compare each term in the orthonormal sequence with the corresponding Legendre polynomial term, remembering the normalization coefficient in the o.n. sequence and the \( \sqrt{n + 1/2} \) term in \( P_n(x) \).

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Most popular questions from this chapter

Show that a non-zero vector \(u\) is an cigenvector of an operator \(A\) if and only if \(|\langle u \mid A u\rangle|=\| A u|| u \mid .\)

If \(A\) is a symmetric operator, show that \(A^{*}\) is symmetric if and only if it is self-edjoint, \(A^{*}=A^{* *}\)

An operator \(A\) is called nermal if it is bounded and commutes with its adjoint. \(A^{*} A=A A^{*} .\) Show that the operator $$ A \psi(x)=c \psi(x)+l \int_{a}^{t} K(x, y) \psi(y) \mathrm{d} y $$ on \(L^{2}([a, b])\), where \(c\) is a real number and \(K(x, y)=\overline{K(y, x)}\), is normal. (a) Show that an operator \(A\) is normal if and only if \(\|A u\|=\left\|A^{*} u\right\|\) for all vectors \(u \in \mathcal{H}\). (b) Show that if \(A\) and \(B\) are commuting normal operators, \(A B\) and \(A+\lambda B\) are normal for all \(\lambda \in \mathbb{C}\)

Let \(A\) be a bounded openitor on a Hilbert space \(\mathcal{H}\) with a one- damensional rangc. (a) Show that there exist vectors \(u, v\) such that \(A x=\langle v \mid x\rangle u\) for all \(x \in \mathcal{H}\). (b) Show that \(A^{2}=\lambda A\) for some scalar \(\lambda\), and that \(\|A\|=\|u\|\|v\|\). (c) Prove that \(A\) is hermitian, \(A^{*}=A\), If and only if there exists a real number \(a\) such that \(v=a u\).

For unbounded operators, show that \(A^{*}+B^{*} \subseteq(A+B)^{\circ}\)
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