Lab 2: Heisenberg's Uncertainty Principle (PHY290) PDF

8 LAB 2 HEISENBERG’S UNCERTAINTY PRINCIPLE (PHY290) APPARATUS Laser, He-Ne, =632.8 nm Diaphragm, 2 single slits, 0.1 mm, 0.2 mm Diaphragm holder Photoelement Universal measuring amplifier Optical profile-bench, 1500 mm Base opt. profile-bench, adjust. Slide mount opt. profi...

8 LAB 2 HEISENBERG’S UNCERTAINTY PRINCIPLE (PHY290) APPARATUS Laser, He-Ne, =632.8 nm Diaphragm, 2 single slits, 0.1 mm, 0.2 mm Diaphragm holder Photoelement Universal measuring amplifier Optical profile-bench, 1500 mm Base opt. profile-bench, adjust. Slide mount opt. profile-bench, h 80 mm BACKGROUND POSITION AND MOMENTUM OF LIGHT A laser beam is a directed beam of light containing many photons. Each photon in the laser beam must obey the rules of quantum mechanics, including Heisenberg’s Uncertainty Principle, which states that it is impossible to know both the position and momentum of a photon at the same time. The photons in the laser beam have a very well-defined momentum. This is why a laser beam can remain fairly small over long distances: we know precisely where the photons are going. Fig. 1 Light passes through a slit of width d and travels a distance L. The diffraction pattern’s central peak has a width Y.  9 When a laser beam gets smaller, like at the focus of a lens, we can pinpoint objects more precisely, since we know the position of the photons much more accurately. However, the beam will expand much more quickly, as we lost accurate knowledge of their momentum. If we have a large laser beam and pass it through a slit, as shown on the image to the right, we restrict the possible positions of the photons, gaining information about where they were. By doing this, we must lose information about their momentum (where they were going). After the slit, the different possible directions (momenta) split up and form a diffraction pattern on a screen, following the distribution seen on the left. The central spot size Y can be related to the uncertainty in momentum. Before passing through the slit, the photons’ momentum was entirely perpendicular to the slit. After the slit, the photons may have a component both parallel and perpendicular to the slit. This change in the possible photon momenta is a result of the uncertainty principle. By passing through the slits, the photon position is known to be within ∆𝑦 = 𝑑. According to the uncertainty principle, this results in an increase in the photon momentum uncertainty as: ∆𝑦 ∙ ∆𝑃 ≥ ℎ (1) whereℎ = 6.626 × 10−34 𝐽𝑠 is Planck’s constant. Fig. 2 Graphical representation of Heisenberg’s uncertainty principle. As we gain certainty in the position “x” we lose it in momentum “p”.  10 By decreasing the uncertainty in the location of the photons within the slit, we’re increasing the uncertainty of the spread in their momenta. This results in the intensity pattern’s width spreading as Y gets smaller. This effect is only significant when the width of the slit is comparable to the wavelength of the light used. The momentum of a photon is given by its wavelength (color) as: ℎ 𝑃= 𝜆 (2) Therefore, blue photons have a much higher momentum than the red photons. If we change the color of the laser, we should see a change in the width of the pattern. OBSERVATION FROM THE WAVE PATTERN VIEWPOINT When a parallel, monochromatic and coherent light beam of wave-length 𝜆 passes through a single slit of width d, a diffraction pattern with a principal maximum and several secondary maxima appears on the screen (Fig. 3). Fig. 3 Diffraction (Fraunhofer) at great distance (Sp = aperture or slit, S = screen) The intensity, as a function of the angle of deviation 𝛼, in accordance with Kirchhoff’s diffraction formula, is sin 𝛽 2 𝐼(𝛼) = 𝐼(0)( 𝛽 ) (3) 𝜋𝑑 where 𝛽 = 𝜆 sin 𝛼 𝜆 The intensity minima are at 𝛼𝑛 = 𝑎𝑟𝑐 sin 𝑛 𝑑 where n = 1, 2, 3 …  11 The angle for the intensity maxima are 𝜆 𝜆 𝛼0 = 0 , 𝛼1 = 𝑎𝑟𝑐 sin 1.430 , 𝛼2 = 𝑎𝑟𝑐 sin 2.459 𝑑 𝑑 The relative heights of the secondary maxima are: 𝐼(𝛼1 ) = 0.0472 𝐼(0) , 𝐼(𝛼2 ) = 0.0165 𝐼(0) (4) QUANTUM MECHANICS TREATMENT The Heisenberg uncertainty principle states that two canonically conjugate quantities such as position and momentum cannot be determined accurately at the same time. Let us consider, for example, a totality of photons whose residence probability is described by the function ƒy and whose momentum by the function ƒp. The uncertainty of location y and of momentum p are defined by the standard deviations as follows eq. (1) the equals sign applying to variables with a Gaussian distribution. For a photon train passing through a slit of width d, the expression is ∆𝑦 = 𝑑 (5) Whereas the photons in front of the slit move only in the direction perpendicular to the plane of the slit (x-direction), after passing through the slit they have also a component in the y-direction. The probability density for the velocity component vy is given by the intensity distribution in the diffraction pattern. We use the first minimum to define the uncertainty of velocity (Figs. 3 and 4). ∆𝑣𝑦 = 𝑐 sin 𝛼1 (6) where 𝛼1 = angle of the first minimum. The uncertainty of momentum is therefore ∆𝑝𝑦 = 𝑚𝑐 sin 𝛼1 (7) where m is the mass of the photon and c is the velocity of light. The momentum and wavelength of a particle are linked through the de Broglie relationship: ℎ 𝜆 = 𝑝 = 𝑚𝑐 (8) Thus, ℎ ∆𝑝𝑦 = sin 𝛼1 𝜆 (9)  12 Fig. 4 Geometry of diffraction at a single slit a) path covered b) velocity compoment of a photon The angle 𝛼1 of the first minimum is thus 𝜆 sin 𝛼1 = 𝑑 (10) according to (3). If we substitute (10) in (9) and (5) we obtain the uncertainty relationship Δ𝑦∆𝑝𝑦 = ℎ (11) If the slit width ∆𝑦 is smaller, the first minimum of the diffraction pattern occurs at larger angles 𝛼1. In our experiment the angle 𝛼1 is obtained from the position of the first minimum (Fig. 4a): 𝑎 tan 𝛼1 = 𝑏 (12) If we substitute (12) in (9) we obtain ℎ 𝑎 ∆𝑝𝑦 = sin(arc tan ) 𝜆 𝑏 (13) Substituting (5) and (13) in (11) gives 𝑑 𝑎 𝜆 sin (arc tan ) = 1 𝑏 (14) after dividing by h.  13 SETUP AND PROCEDURE 1. Different screens with slits (0.1 mm and 0.2 mm) are placed in the laser beam one after the other. The distribution of the intesnity in the diffraction pattern is measured with the photo-cell as far behind the slit as possible. A slit (0.3 mm wide) is fitted in front of the photocell (Fig. 5). Fig. 5 Experimental set-up for measuring the distribution of intensity in diffraction patterns. 2. The voltage drop at the resistor attached parallel to the imput of the universal measuring amplifier is measured and is approximately proportional to the intensity of the incident light. 3. The principal maximum, and the first secondary maximum on one side, of the symmetrical diffraction pattern of a slit 0.1 mm wide (for example) are recorded. 4. The intensity in the diffraction pattern from the photocurrent is plotted as a function of the position. (Fig. 6) 5. For the other slits, it is sufficient to record the two minima to the right and left of the principal maximum, in order to determine a  14 Fig. 6 Intensity in the diffraction pattern of a 0.1 mm wide slit at a distance of 1140 mm. The photocurrent is plotted as a function of the position. PART I: THE FRAUNHOFER DIFFRACTION PATTERN To measure the intensity distribution of the Fraunhofer diffraction pattern of a single slit (0.1 mm and 0.2 mm). The heights (𝑰(𝜶𝒏 )) of the maxima (eq. 4) and the positions (𝜶𝒏 ) of the maxima and minima are calculated according to Kirchhoff’s diffraction formula and compared with the measured values. PART II: HEISENBERG’S UNCERTAINTY PRINCIPLE To calculate the uncertainty of momentum from the diffraction patterns of single slits of differing widths (0.1 mm and 0.2 mm) and to confirm Heisenberg’s uncertainty principle. (eq. 14) Important: In order to ensure that the intensity of the light from the laser is constant, the laser should be switched on about half an hour before the experiment is due to start. The measurements should be taken in a darkened room or in constant natural light. If this is not possible, a longish tube about 4 cm in diameter and blackened on the inside (such as a cardboard tube used to protect postal packages) can be placed in front of the photcell. Caution: Never look directly into a non attenuated laser beam