130_notes.dvi

xis|f(x)|^2 dx. The probability for a particle to have wave number in regiondkaround some value ofkis|A(k)|^2 dk. (Remember that ...

also a Gaussian. We will show later that a Gaussian is the best one can do to localize a particle in position and momentum at th ...

∆p∆x≥ ̄h 2 It says we cannot know the position of a particle and its momentum atthe same time and tells us the limit of how well ...

Our Fourier Transform can now be read to say that weadd up states of definite momentum to getψ(x) ψ(x) = ∫∞ −∞ φ(p)up(x)dp and w ...

To cover the general case, lets expandω(k) around the center of the wave packet in k-space. ω(k) =ω(k 0 ) + dω dk ∣ ∣ ∣ ∣ k 0 (k ...

so that we can easily compute the coefficients. an= 1 2 L ∫L −L f(x)e −inπx L dx In summary, the Fourier series equations we wil ...

This is just the extension of the Fourier series to allx. Iff(x) is normalized, thenA(k) will also be normalized with this (symm ...

5.6.4 Fourier Transform of Gaussian* We wish toFourier transform the Gaussian wave packetin (momentum) k-spaceA(k) = ( 2 α π ) 1 ...

Letscheck the normalization. ∫∞ −∞ |f(x)|^2 dx= √ 1 2 πα ∫∞ −∞ e− x 22 α dx= √ 1 2 πα √ 2 απ= 1 Given a normalizedA(k), we get a ...

We write explicitly thatwdepends onk. For our free particle, this just means that the energy depends on the momentum. To cover t ...

Use the fine structure constant to avoid CGS units which are used inthe textbook. α= e^2 ̄hc = 1/ 137 This combination saves a l ...

5.7 Examples 5.7.1 The Square Wave Packet Given the following one dimensional probability amplitude in the position variable x, ...

5.7.3 The Dirac Delta Function Wave Packet* Given the following one dimensional probability amplitude in the position variable x ...

5.7.6 Estimate the Hydrogen Ground State Energy The reason the Hydrogen atom (and other atoms) is so large is the essentially un ...

5.8 Sample Test Problems A nucleus has a radius of 4 Fermis. Use the uncertainty principle toestimate the kinetic energy for a ...

aψ(x) =√^12 afor−a < x < a, otherwiseψ(x) = 0. bψ(x) = (απ) (^14) e−αx (^2) / 2 c ψ(x) =δ(x−x 0 ) Use the Heisenberg unce ...

6 Operators Operators will be used to help us derive a differential equation that our wave-functions must satisfy. They will als ...

6.1.2 The Energy Operator We can deduce and verify (See section 6.6.2) theenergy operatorin the same way. E(op)=i ̄h ∂ ∂t 6.1.3 ...

and p(op)=p. The (op) notation used above is usually dropped. If we see the variablep, use of the operator is implied (except in ...

The expectation values of physical quantities should be real. Gasiorowicz Chapter 3 Griffiths Chapter 1 Cohen-Tannoudji et al. C ...