Phy107Spr06Lect25

Phy107Spr06Lect25 - N o o ffic e h o u rs T u e sd a y G u...

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1 Mon. Mar. 27, 2006 Phy107 Lecture 25 1 From Last Time… • Particle can exist in different quantum states, having — Different energy — Different momentum — Different wavelength • The quantum wavefunction describes wave nature of particle. • Square of the wavefunction gives probability of finding particle. • Zero’s in probability arise from interference of the particle wave with itself. No office hours Tuesday Mon. Mar. 27, 2006 Phy107 Lecture 25 2 Particle in a box Lowest energy state Next higher energy state 3rd energy state Wavefunction Probability L Mon. Mar. 27, 2006 Phy107 Lecture 25 3 Wavefunction of pendulum n =2 n =3 n =1 ground state Here are quantum wavefunctions of a pendulum. Which has the lowest energy? Mon. Mar. 27, 2006 Phy107 Lecture 25 4 Classical vs quantum Low classical amplitude, low energy Higher classical amplitude, higher energy Mon. Mar. 27, 2006 Phy107 Lecture 25 5 Probability density of oscillator Moves fast here, low prob of finding in a ‘blind’ measurement Moves slow here, high prob of finding Classical prob Mon. Mar. 27, 2006 Phy107 Lecture 25 6 Wavefunctions in two dimensions • Physical objects often can move in more than one direction (not just one-dimensional) • Could be moving at one speed in x -direction, another speed in y -direction. • From deBroglie relation, wavelength related to momentum in that direction • So wavefunction could have different wavelengths in different directions. " = h p

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2 Mon. Mar. 27, 2006 Phy107 Lecture 25 7 Two-dimensional (2D) particle in box Ground state: same wavelength (longest) in both x and y Need two quantum #’s, one for x -motion one for y -motion Use a pair ( n x , n y ) Ground state: (1,1) Probability (2D) Wavefunction Probability = (Wavefunction) 2 One-dimensional (1D) case Mon. Mar. 27, 2006 Phy107 Lecture 25 8 2D excited states ( n x , n y ) = (2,1) ( n x , n y ) = (1,2) These have exactly the same energy, but the probabilities look different. The different states correspond to ball bouncing in x or in y direction. Mon. Mar. 27, 2006 Phy107 Lecture 25 9 The classical version Same velocity (energy), but details of motion are different. Motion in x direction Motion in y direction Mon. Mar. 27, 2006 Phy107 Lecture 25 10 Particle in a box What quantum state could this be? A. n x =2, n y =2 B. n x =3, n y =2 C. n x =1, n y =2 Mon. Mar. 27, 2006 Phy107 Lecture 25 11 Next higher energy state • The ball now has same bouncing motion in both x and in y. • This is higher energy that having motion only in x or only in y . ( n x , n y ) = (2,2) Mon. Mar. 27, 2006 Phy107 Lecture 25 12 Three dimensions • Object can have different velocity (hence wavelength) in x , y , or z directions. – Need three quantum numbers to label state
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Phy107Spr06Lect25 - N o o ffic e h o u rs T u e sd a y G u...

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