uncertainty

THE UNCERTAINTY PRINCIPLE

Cruz. Dy. EspirituPhysics Reporting

4th Quarter


Name: Werner Heisenberg

(1901-1976)

+Made many significant

contributions to physics, like the

Uncertainty Principle (this won the

Nobel Prize in 1932).

+Developed an abstract model of

quantum mechanics called matrix

mechanics.

+Predicted two forms of molecular

hydrogen, and theoretical models of

the nucleus.


• If you were to measure the position and speed of a particle

at any instant, you would always be faced with

experimental uncertainties in your experiments.

• Based on classical mechanics, no fundamental barrier to an

ultimate refinement of the apparatus or experimental

procedure exists. This means that it is possible, in

principle, to make such measurements with arbitrarily small

uncertainty. Quantum theory however, predicts that such a

barrier exists. This is best explained by the Heisenberg

uncertainty principle.


• If a measurement of position is made with precision dx and

a simultaneous measurement of linear momentum is made

with precision dpx, then the product of the two

uncertainties can never be smaller than h/2

where h=h/2pi.

Thus, it is physically impossible to measure simultaneously

the exact position and exact linear momentum of a particle, due to

the inverse relationship between dx and dpx.


• This stems not from imperfections in measuring instruments, but rather from the quantum structure of matter---

From effects such as the unpredictable recoil of an electron when struck by a photon or the diffraction of light or electrons through a slit.


• Here's a thought experiment:

Suppose you wanted to measure the position and linear momentum of an electron as accurately as possible. You might be able to do this by viewing the electron with a powerful light microscope. For you to be able to see the electron and thus determine its location, at least one photon of light must bounce off the electron, and pass through the microscope into your eye.

But when it strikes the electron, the photon imparts some unknown amount of its momentum to the electron. Thus, in the process of your locating the electron very accurately, that is, making dx very small by using a light with short wavelength (which has high momentum)---the very light that allows you to succeed changes the electron's momentum to some undeterminable extent (making dpx very great).


• In analyzing the collision, note that the incoming photon has momentum h/pi. As a

result of the collision, the photon transfers part of all of its momentum along the x-

axis to the electron. Thus, the uncertainty in the electron's momentum after the

collision is as great as the momentum of the incoming photon: dpx = h/pi.

• Furthermore, since the photon also has wave properties, we expect to be able to

determine its position to within one wavelength of the light being used to view it,

so dx = lambda.

• Multiplying these two uncertainties gives: dx * dpx = lambda (h/lambda) = h. The

value h represents the minimum in the products of the uncertainties. Because the

uncertainty can always be greater than this minimum, we have: dx * dpx >= h.

Apart from the numerical factor 1/4pi introduced by Heisenberg's more precise

analysis, this agrees with


• In summary, the Heisenberg Uncertainty Principle is applied such that the better you know that position of a particle, the less you know about its momentum. This goes vice versa. To put it into a equation,

• Dx is the measurement uncertainty in the particle's x position. Dpx is its measurement uncertainty in its momentum (recall: mass*velocity or kg*m/s) in the x direction and

• This relation holds true for all three dimensions. Therefore:

References

• John Wiley &Sons Inc. (2012). Quantum Physics and the Heisenberg Uncertainty Principle. Retrieved from http://www.dummies.com/how-to/content/quantum-physics-and-the-heisenberg-uncertainty-pri.html

• Serway, R.A. & Beichner, R.J. (1982). Physics For Scientists and Engineers with Modern Physics 5th Edition. Saunders College Publishing: Florida, Orlando.

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