9783642310591-c1.pdf - Chapter 1 Recollections from...

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Chapter 1 Recollections from Elementary Quantum Physics Abstract We recall the prerequisites that we assume the reader to be familiar with, namely the Schrödinger equation in its time dependent and time independent form, the uncertainty relations, and the basic properties of angular momentum. Introductory courses on quantum physics discuss the one-dimensional Schrödinger equation for the wave function Ψ (x,t) of a particle of mass M moving in a poten- tial V i ∂Ψ ∂t = − 2 2 M 2 Ψ ∂x 2 + V Ψ. (1.1) Therein = h/ 2 π is the reduced Planck constant. The function Ψ is understood as a probability amplitude whose absolute square | Ψ (x,t) | 2 = Ψ (x,t)Ψ (x,t) gives the probability density for finding the particle at time t at position x . This probability density is insensitive to a phase factor e i ϕ . With the Hamilton operator H = − 2 2 M 2 ∂x 2 + V, (1.2) the Schrödinger equation reads ˙ Ψ = − i H Ψ. (1.3) The dot denotes the time derivative. In this text, we print operators and matrices in nonitalic type, like H, p , or σ , just to remind the reader that a simple letter may represent a mathematical object more complicated than a number or a function. Ordinary vectors in three-dimensional space are written in bold italic type, like x or B . The time dependent Schrödinger equation reminds us of the law of energy con- servation E = p 2 / 2 M + V if we associate the operator i ∂/∂t with energy E and the operator ( / i )∂/∂x with momentum p . Take as an example the plane wave Ψ (x,t) = Ψ 0 e i (kx ωt) of a photon propagating in the x -direction. The energy opera- tion i ∂Ψ/∂t = ωΨ then relates energy to frequency, E = ω , and the momentum operation ( / i )∂Ψ/∂x = relates momentum to wave number, p = k . D. Dubbers, H.-J. Stöckmann, Quantum Physics: The Bottom-Up Approach , Graduate Texts in Physics, DOI 10.1007/978-3-642-31060-7_1 , © Springer-Verlag Berlin Heidelberg 2013 3
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4 1 Recollections from Elementary Quantum Physics The probability amplitude Ψ (x,t) for finding the particle at time t at position x and the amplitude Φ(p,E) for finding it with energy E and momentum p turn out to be Fourier transforms of each other. Pairs of Fourier transforms have widths that are reciprocal to each other. If the width x in position is large, the width p in momentum is small, and vice versa, and the same for the widths t and E . The conjugate observables p and x or E and t obey the uncertainty relations p x 1 2 , E t 1 2 . (1.4) The exact meaning of x , etc. will be defined in Sect. 3.4. For stationary , that is, time independent potentials V (x) , the Hamilton operator ( 1.2 ) acts only on the position variable x . The solution of the Schrödinger equation then is separable in x and t , Ψ (x,t) = ψ(x) e i Et/ . (1.5) The probability density (in units of m 1 ) for finding a particle at position x then is independent of time | Ψ (x,t) | 2 = | ψ(x) | 2 . The amplitude ψ(x) is a solution of the time independent Schrödinger equation 2 2 M 2 ψ(x) ∂x 2 + V (x)ψ(x) = Eψ(x).
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