physics-subatomic-particles

can decrease the illumination to the absolute minimum : one photon . Then, when thi sphoton hits the stone, it will cause it to change its course . So, let us now try anddecrease the energy of the photon . To do this, we must increase its wavelength, bu tthis blurs the image in the came-a . So it seems that we have a limit to our accurac yin measurement .Let us now consider the illustration which Heisenberg himself gave . We wish t olocate an electron by means of gamma rays . It is known that the error in determinin gthe position of an object by means of a microscope is in the order of the wavelengthof the radiation used . In fact, two points can not be resolved or recognised a sdistinct by means of a lens if they are closer than X /A to each other, where \ is th ewavelength of the radiation used, and A is the angle subtended at the lens by eithe rof the two points . The error in the determination of the position of the electro nwe say to be s, and we know that ' s --WA . But the rays which fall on the electro ngive it momentum, according to the Compton effect, and each photon has a momentum o fby/c, where v is its frequency, c is the velocity of light in vacuo, and h is Planck' sconstant . But our photon could have come from any part of the lens, and so th euncertainty in its vector is in the order of the width of the lens, This distance ,because angle A is &mall, is in the order of by/c, the momentum of the electron, time sangle A . Thus we may say that E p -- hvA/c, where p is the momentum of the electron ,and thereforepbs me, h .This example brings out the point that the uncertainty in a measurement arises fromthe disturbance introduced in the process of measurement .Before we discuss the Uncertainty Principle, as it is known, in any more detail ,let us first discuss two more methods of arriving at it . The first is as follows .We shall consider the problem of the accurate measurement of position and momentum i na one dimensional world, where there exist units such that X e 1 . This means that i nour world, p = 2e/a, and thus the accuracy with which we may measure the wavelengthof the de Broglie wave of this particle is linearly related to the accuracy with whic hwe may measure its momentum . Let us assume that the wave of this particle is a sine oneof finite length . The more full cycles of the sine we may observe, the more accur at elywe may measure its wavelength, and hence the momentum of our particle . However, th emore cycles we observe, the less accurately we may find the position of our particl eat any one moment . Thus, if we say that the uncertainty in measuring the position, x ,of our particle is b x, then we may say that x 'A, where a is the wavelength o four sine curve, and n is the number of full cycles we may observe . Furthermore, weknow that 1/n ,D/), , where S)' is the uncertainty in measurement of the wavelength .In our world where X . 1, we know that e 20/p, and so . EX/ Sp/p, and n>'' 2rtn/p .Combining our equations, and returning to the normal system of measurements, we findthat 3 x 8'p h . However, we have here considered a particle whose wave is a sin ecurve, the simplest and most regular of al] . wave functions . Thus, in a real situation ,we may say thatbpc~x h ,because all real curves will have wavelengths which are much more difficult to measure .Generalising, we obtain the first set of so-called 'uncertainty relations' :Es„ by >, h, bsg 3p ~ . h, and bs= Sp >i h .Our second method is simply to take an analogy from classical wave theory, and appl yit, with modifications, to de Broglie waves . Suppose that two coincident sources o fsound emit notes of frequency v and Sv respectively, Zv being small compared with v .