THE UNCERTAINTY PRINCIPLE SHINTARO FUSHIDA-HARDY 1. Heisenberg uncertainty principle Suppose p : R ! R is a probability density function for a random variable X. The moments of X are given by Z n n (1) mn := E(X ) = x p(x)dx: R 2 (Assuming they exist), the first moment mn is the mean of X, while m2 −m1 is the variance of X. The Heisenberg uncertainty principle in quantum mechanics states that 2 (2) Var(x)Var(p) ≥ ~ 4 where x and p denote position and momentum of a particle, respectively. If these both have mean zero, the above equation is Z Z 2 2 2 ~ (3) t px(t)dt s pp(s)dx ≥ ; R R 4 where px and pp are probability density for position and momentum. But from quantum mechanics, the probability density for position is precisely the wave function of the particle under consideration. Moreover, pp is the Fourier transform of px with additional constants. In this document we prove a more general result from which the Heisenberg uncertainty principle follows. 2. Definitions and Theorems The Schwartz space on R are is the collection of rapidly decreasing complex valued smooth functions on R. Formally, 1 (4) S(R) = ff 2 C (R): kfkα,β < 1g; where the collection of norms is defined for all α; β 2 N by α (β) (5) kfkα,β = sup jx f (x)j: x2R The Schwartz space is the natural space for Fourier analysis, since the Fourier transform is a homeomorphism on this space. It also has the delicious property of being contained in p L (R) for all 1 ≤ p ≤ 1. The Fourier transform is defined by Z (6) f^(ζ) = f(x)e−2πixζ dx: R 1 On S(R), the inverse is simply Z (7) f(x) = f^(ζ)e2πixζ dζ: R Moreover, the Parseval identity holds for all f; g 2 S(R): Z Z (8) f(x)g^(x)dx = f^(x)g(x)dx: R R The Plancherel identity immediately follows: Z Z (9) jf(x)j2dx = jf^(x)j2dx: R R Lemma 2.1. Let f 2 S(R). Then f^0(ζ) = 2πiζf^(ζ). Proof. This follows from integration by parts: Z h i1 Z 2πiζf^(ζ) = 2πiζ f(x)e−2πixζ dx = − f(x)e−2πixζ + f 0(x)e−2πixζ dx: −∞ R R The first term on the right hand side vanishes since f is a Schwartz function. 3. Fourier uncertainty principle Recalling the form of the uncertainty principle in equation 2, we prove the following Fourier uncertainty principle: Theorem 3.1. Let f 2 S(R). Suppose f is normalised, that is, kfk2 = 1. Then Z Z 2 2 2 ^ 2 1 (10) x jf(x)j dx ζ jf(ζ)j dζ ≥ 2 : R R 16π Proof. Let f 2 S(R). Integration by parts gives Z h i1 Z xf(x)f 0(x)dx = xf(x)f(x) − (xf(x))0f(x)dx −∞ R R Z Z = 0 − f(x)f(x)dx − xf 0(x)f(x)dx: R R The first term on the right side vanishes by virtue of f being a Schwartz function. Rear- ranging the above gives Z Z 1 = jf(x)j2dx = − x(f 0(x)f(x) + f(x)f 0(x))dx R R Z = −2 xReal[f(x)f 0(x)]dx: R By the Cauchy-Schwartz inequality, Z 2 Z Z 1 ≤ 2 jxReal[f(x)f 0(x)]jdx ≤ 4 jxf(x)j2dx jf 0(x)j2dx : R R R 2 Moreover, by lemma 2.1 and the Plancherel identity, Z Z Z Z (11) jf 0(x)j2dx = jf^0(ζ)j2dζ = j2πiζf^(ζ)j2dζ = 4π2 jζf^(ζ)j2dζ: R R R R Combining this with the previous inequality, Z Z (12) 1 ≤ 16π2 jxf(x)j2dx jζf^(ζ)j2dζ ; R R so Z Z 2 2 2 ^ 2 1 (13) x jf(x)j dx ζ jf(ζ)j dζ ≥ 2 R R 16π as required. Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand Email address: [email protected] 3.