Feynman Lectures Simplified 4A: Math for Physicists by Robert Piccioni

Author:Robert Piccioni
Language: eng
Format: epub
Tags: Calculus, Advanced Algebra, Probability & Statistics, Tensors & Matrices, Infinite Series, Vector Algebra, Fourier Series
Publisher: Real Science Publishing
Published: 0101-01-01T00:00:00+00:00

This confirms the the normalization factor.

Now let’s calculate the variance.

σ2 = ∫ x2 Prob(x) dx

Let’s begin by defining W:

W = σ2 √(2π) = ∫–∞+∞ x2 exp{–x2/2} dx

Choose u and v such that:

v = x

dv = dx

u = –exp{–x2/2}

du = +x exp{–x2/2}

We now integrate W by parts.

W = ∫ v [du] = v [u] – ∫ u dv

W = ∫–∞+∞ x [x exp{–x2/2}] dx

W = – x exp{–x2/2} |–∞+∞ + ∫ exp{–x2/2} dx

W = 0 + √(2π)

σ2 = W /√(2π) = 1

QED

Completing the Square

Another trick applies to integrals like:

Q = ∫ exp{x2+bx} dx

Here, b is either a constant or is not a function of x.

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