PHY 1371Dr. Jie Zou1 Chapter 41 Quantum Mechanics

PHY 1371Dr. Jie Zou2 Outline The double-slit experiment revisited The uncertainty principle

PHY 1371Dr. Jie Zou3 The double-slit experiment produced by electrons Wave-particle duality of material particles Consider the diffraction of electrons passing through a double slit. Results: If the detector detects electrons at different positions for a sufficiently long period of time, one finds an interference pattern representing the number of electrons arriving at any positions along the detector line. Such an interference pattern cannot occur if electrons behave as classical particles, and hence electrons are behaving as waves.

PHY 1371Dr. Jie Zou4 A detailed look at the double- slit experiment (a)-(c): Computer simulation. (d): Real photograph of a double-slit interference pattern produced by electrons The experiment is carried out at a low beam intensity over long exposure. After a short exposure (after 28 electrons ): Individual blips hitting in an apparently random pattern. After long exposure (after 10000eelctrons): Interference pattern becomes clearer.

PHY 1371Dr. Jie Zou5 Significance of the double-slit experiment of electrons The wave-particle dual nature of electrons is clearly shown in the experiment: Although the electrons are detected as particles at a localized spot at some instant of time, the probability of arrival at that spot is determined by the intensity of two interfering matter waves. Interpretation of matter waves (first suggested by Max Born in 1928): In quantum mechanics, matter waves are described by the complex-valued wave function . The absolute square |  | 2 =  *  : |  | 2 gives the probability of finding a particle at a given point at some instant. The wave function  contains all the information that can be known about the particle.

PHY 1371Dr. Jie Zou6 Interference of matter waves With only slit 1 open, the probability of detecting the electron at the detector is given by |  1 | 2 (similarly for |  2 | 2 ). With both slits open, the electron is in a superposition state:  =  1 +  2. Probability of detecting the electron at the detector: |  | 2 = |  1 +  2 | 2 = |  1 | 2 + |  2 | 2 +2 |  1 ||  2 | cos .  = the relative phase difference between  1 and  2 at the detector. An electron’s wave property interacts with both slits simultaneously. The electron passes through both slits. (a): Blue curve (b): Blue curve (c): Red curve

PHY 1371Dr. Jie Zou7 Example Problem #1 Neutrons traveling at m/s are directed through a double slit having a 1.00-mm separation. An array of detectors is placed 10.0 m from the slit. Neutron mass m n = x kg. (a) What is the de Broglie wavelength of the neutrons? (b) How far off axis is the first zero-intensity point on the detector array? (c) When a neutron reaches a detector, can we say which slit the neutron passed through? Explain.

PHY 1371Dr. Jie Zou8 The uncertainty principle Heisenberg uncertainty principle: If a measurement of position is made with precision  x and a simultaneous measurement of linear momentum is made with precision  p x, then the product of the two uncertainties can never be smaller than ħ/2:  x  p x >= ħ/2, where ħ = h/2 . It is impossible to measure simultaneously the exact position and exact linear momentum of a particle. The inescapable uncertainties  x and  p x do not arise from the imperfections in measuring instruments. Rather, they arise from the quantum structure of matter.

PHY 1371Dr. Jie Zou9 Example 41.2 Locating the electron The speed of an electron is measured to be 5.00 x 10 3 m/s to an accuracy of %. Find the minimum uncertainty in determining the position of this electron.

PHY 1371Dr. Jie Zou10 Homework Ch. 40, P. 1317, Problems: #46, 51.