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Unformatted text preview: Also we have Equations (I) and (II) are basic formulations of the uncertainty principle. e.g. the product ! p x ! x cannot be inFnitesimally small but must equal in magnitude at least Planck's constant, h. Likewise for ! E ! t. E = h ! " ! E = h ! ! since ! t ! ! # 1 " ! E = h ! ! # h ! t " ! E ! t # h (II) In text books sometimes see: ! p ! x " h 2 = h 4 # ! E ! t " h 2 This comes about from a more precise determination. These fundamental limitations show that if we use a particle description for electromagnetic (e.m.) radiation or a particle to explain a phenomenon then the wave aspect is suppressed and vice versa. ! p x " h ! x = # For ! t = 0 then: ! E ! t " h => ! E = and since ! E = h ! ! ! !> ! ! and therefore the wave aspect is suppressed since all the quantities ! , E, " and p ( )) > # ) are completely uncertain. For example, if the particle nature of say an electron is to be displayed, both ! x and ! t must be zero: i.e ! x = 0 then: ! p x ! x " h Also we have ! p x = h " 2 !" so !" = # . Alternatively, if the wave characteristics of a particle or e.m. radiation are to be deFned perfectly then ! ! = 0 and ! " = 0 (so that ! E = 0 and ! p = 0) and the particle characteristics of precise location in space and time are completely uncertain. Therefore, as Frst pointed out by Bohr , the waveparticle duality is complementary rather than contradictory. As different as they are, both wave and particle aspects are required and they complement each other to fully describe matter or electromagnetic radiation. The uncertainty principle is a direct consequence of the wave character of moving bodies (particles) and has nothing to do with experimental uncertainties or how good the measuring equipment is! Examples 1. electron v = 300 m/sec ! v = 0.010% i.e. velocity accurate to 0.010% p = mv = (9.11 x 1031 kg) x 300 m/sec = 2.7 x 1028 kg m/sec & ! uncertainty in momentum ! P is also 0.010% since ! p = m ! v ! ! p = 0.010 100 x 2.7 x 1028 kg.m sec = 2.7 x 1032 kg m/sec Hence the minimum uncertainty in position is _______________________________ ! x = h ! p = 6.6 x 1034 J.S 2.7 x 1032 kg m/s = 2.4 cm. 2. Bullet v = 300 m/sec m = 50 gm ! v= 0.010% p = mv = 15 kg m/sec. ! P = ¡¢¡£¡ £¡¡ x £¤ KG¢M SEC = £¢¤ x £¡¥ KG¢M¦SEC hEnCE ! x = 6¢6 x £¡¥4 J¢§¢ £¢¤ x £¡¥ KG¢M¦SEC = 4¢4 x £¡¥£ M¢ This is beyond possibility of measurement! This is beyond possibility of measurement! So if the bullets velocity is known to within 0.010% then we can determine exactly where the bullet is! h 2 2 m " 2 " x 2 # $ % & ’ ( ) ( x , y , z ) + V ( x ) ) ( x ) = E ) ( x ) Schrödinger Wave equation 1925 In order to account for the wave nature of particles, a new mechanics was developed (wave mechanics) to give the following equation of motion: This is called the time independent Schrödinger Equation and is the equation of motion for systems with quantum characteristics. In wave mechanics it replaces Newton’s Laws (i.e. )....
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This note was uploaded on 08/20/2010 for the course PHYS 142 taught by Professor Xxx during the One '09 term at University of Wollongong, Australia.
 One '09
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