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Ohanian’s Book – Part1

Posted September 8, 2012
Filed under: 3. Ohanian's Book - Part 1, D. PROOFS |
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A relatively recent (1990) textbook by Ohanian [1] gives the proof of uncertainty principle in a different way. However, it is still based on Fourier Transform (FT), that Heisenberg used, which we have described in another section.  In this part we show how FT is introduced in the proof, relating position and momentum. In the second part we actually derive the uncertainty relation.

Ohanian’s proof is distributed over many chapters and pages in his book.  He begins his chapter two, on page 21, with the following physical relations (1) obtained from experiments and calculations

 \ \upsilon =\frac{E}{h} , \lambda =\frac{h}{p} , k=\frac{2\pi }{\lambda }=\frac{p}{h} , w=2\pi \upsilon , \bar{h}=\frac{h}{2\pi }

(1)

Then on page 22, Ohanian starts with four conceivable harmonic waves (2)

 \ e^{ikx-iwt} , e^{-ikx+iwt} , \sin (kx-wt), and \cos (kx-wt)

(2)

Using some selection criteria he finally decides that  \ e^{ikx-iwt} from the list in (2) as the correct wave function describing a free particle. This wave function in a sense defines the position of the particle, so we may call it also the position function. Then he finds the differential equation that this function must satisfy, using the equivalent principle we discussed. He writes, it is easy to check that the second derivative of  \ e^{ikx-iwt} with respect to x and the first derivative with respect to time t are proportional:

 \ -\frac{\bar{h}^{2}}{2m}\frac{\partial ^{2}}{\partial x^{2}} e^{ikx-iwt}=i\bar{h} \frac{\partial }{\partial t} e^{ikx-iwt}

(3)

The equation (3) is a linear equation and therefore will be compatible with the superposition principle. So any function  \ \psi (x,t)  that can be written as a superposition of a finite or infinite number of waves of the type  \ e^{ikx-iwt} will also satisfy the differential equation (4)

 \ -\frac{\bar{h}^{2}}{2m}\frac{\partial ^{2}}{\partial x^{2}} \psi (x,t)=i\bar{h} \frac{\partial }{\partial t} \psi (x,t)

(4)

The above (4) is the Schrodinger’s wave equation for a free particle. Then on page 32 Ohanian introduces the inverse FT

 \ \psi (x,t)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{\infty }{\varphi }(k,t)e^{ikx}dk

(5)

Ohanian calls the function  \ \varphi (k,t) as the amplitude in momentum space. We will also call it as the momentum function. Observe how, via inverse FT in (5), the momentum function gets related to the position function. This (5) is purely an assumption. Once such assumption is made, by equivalence principle, it will show up everywhere in all results.

Then he finds  \ \varphi (k,t) by substituting (5) in (4):

 \ -\frac{\bar{h}^{2}}{2m}\lbrack \frac{1}{\sqrt{2\pi }}\int_{-\infty }^{\infty }{-k^{2}\varphi }(k,t)e^{ikx}dk\rbrack =i\bar{h}\lbrack \frac{1}{\sqrt{2\pi }}\int_{-\infty }^{\infty }{ \frac{\partial }{\partial t}\varphi }(k,t)e^{ikx}dk\rbrack

(6)

Now comparing both sides the coefficients of  \ e^{ikx} in (6) we obtain the following differential equation (7)

 \ \frac{\bar{h}^{2}k^{2}}{2m}\varphi (k,t)=i\bar{h} \frac{\partial }{\partial t}\varphi (k,t)

(7)

Which has the solution given by (8)

 \ \varphi (k,t)=\varphi (k,0)e^{-iEt/\bar{h}}

(8)

Where

\ E=\frac{\bar{h}^{2}k^{2}}{2m}

Observe that, we found relations similar to (7) and (8) in the Heisenberg’s proof also. Thus the general solution of the Schrodinger’s wave equation for a free particle can be written from (5) as

 \ \psi (x,t)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{\infty }{\varphi }(k,0)e^{i(kx-\frac{\bar{h}k^{2}}{2m}t)}dk

(9)

The above solution (9) can be written in terms of the momentum \ p=\bar{h}k as in (10)

 \ \psi (x,t)=\frac{1}{\sqrt{2\pi \bar{h}}}\int_{-\infty }^{\infty }{\varphi }(p,0)e^{i(\frac{p}{\bar{h}}x-\frac{p^{2}}{2m\bar{h}}t)}dp

(10)

Thus equation (5) or (10) gives a direct connection of momentum and position via Fourier Transform. Therefore, as mentioned by Heisenberg, the two properties of the particle cannot be independent now. This will naturally dictate the uncertainty principle. The rest of his proof is designed to get the expression for the momentum function. In (8) he got the expression for t dimension. In the next section we will see that he assumes Gaussian function for the k dimension.

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REFERENCES
[1]Ohanian, H.C., Principles of quantum mechanics, Prentice Hall, NJ, 1990.