Chapter. 5 Bound States: Simple Case

Transcription

1 Announcement Course webpage Textbook PHYS-2402 Lecture 12 HW3 (due 3/2) 13, 15, 20, 31, 36, 41, 48, 53, 63, 66 ***** Exam: 3/12 Ch.2, 3, 4, 5 Feb. 26, 2015 Physics Colloquium: today Chapter. 5 Bound States: Simple Case Physics Colloquium Thursday, February 26th at 3:40PM in SC 234 Featuring: Dr. Roy F. Schwitters Department of Physics University of Texas at Austin Purpose: The UT Austin Maya Muon Project: Applying Tools and Methods of High Energy Physics to Archeology Attenuation by energy loss through ionization of naturally occurring cosmic ray muons provides a robust physical mechanism for imaging structure inside large, otherwise inaccessible objects such as archeological ruins, including pyramids, and, even, volcanoes. For the past decade, our group has been adapting tools and methods of high energy physics to developing practical detectors and analysis methods for exploring Mayan ruins in Central America and other archeological sites around the world. Results of performance tests and experience gained in this effort will be described. To make QM useful in real application, we must have a way to account for the effects of external forces** Let s start with the Schrödinger eq. to include these effects. Refreshments at 3:00PM in SC ** interaction of object with its surrounding

2 Outline: Chapter. 5 Bound States: Simple Case The Schrödinger Equation (for Interacting Particles) Stationary States Physics Conditions: Well-Behaved Functions A Review of Classical Bound States (First) Case 1: Particles in a Box The Infinite Well Case 2: The Finite Well Case 3: The Simple Harmonic Oscillator Expectation Values, Uncertainties, and Operators The Schrodinger Equation for Interacting Particles A Particle Interacting With What? F=mg The Schrodinger Equation for Interacting Particles A Particle Interacting With What? Simplification: The Concept of Potential (replaces all individual particle-particle interactions with a single smooth potential) Why? see next page Bound Systems A bound system is any system of interacting particles where the nature of the interactions between the particles keeps their relative separation limited. Classical example: the solar system. In general, the problem is very difficult. Simplification: motion of a single particle that moves in a fixed potential energy field U(x) created by the other particles in the system. A good approximation when the mass of the particle is small compared to the total mass of the system (think heavy nucleus light electron). Classical bound system: E( x) = K( x) + U( x) ( ) = U( x) E x Classically allowed region: ( ) > U( x) E x The probability to find a classical object is proportional to (velocity) -1. The classical probability is peaked at the turning points, where v=0. Classically forbidden region: E x ( ) < U( x) Δx K( x ) > 0

3 The Infinite Square Well - a particle in the potential is completely free, except at the two ends where an infinite force prevents it from escaping Outside the well: ψ ( x) = 0 - the probability of finding the particle =0 2 2 h d ψ Inside the well: = Eψ 2 ( x) 2m dx 2 d ψ ( x) 2 2mE = k ψ 2 ( x) k - the harmonic oscillator equation dx h General solution: ψ x = A kx+ B kx - constants A and B are fixed by boundary conditions ( ) sin cos ( x) Continuity of the wave function: ψ( 0) = ψ( L) = 0 ( ) Thus, ψ ( x) = Asin kx ( L) Asin kl 0 ψ 0 = Asink0+ Bcosk0= B= 0 ψ = = kl = 0, ± π, ± 2 π,... nπ kn =, n= 1,2,... L n quantum number (1D motion is characterized by a single q.n., for 2D motion we need two quantum numbers, etc.) See next time for details Ae i(kx'wt) The Schrodinger Equation for Interacting Particles For free particles or in the absence of external forces Schrödinger eq. is based on E accounting - w/o external interactions Try to add potential energy U(x) Adding P.E. The Schrodinger Equation for Interacting Particles and for Stationary Potentials! Time-dependent Schrödinger Eq.! To determine the behavior of particle in (1) CM: solve F = m(d 2 r/dt 2 ) for r, given knowledge of Net external F on particle in (2) QM: solve the Schrödinger eq. for ψ(x,t), given knowledge of P.E., U(x) U U = U ( x)! U ( t)

4 Key Assumption: Factorization of the wave function Wave function may be express as a product of Standard Math. Technique; Separation of variables and factoring out terms constant w.r.t. the partial derivatives Spatial Part Temporal Part Q: Why?, A: allows us to break a differential eq. with 2 independent variables (x,t) into simpler eqs. For position & time, separately!! What happens with the Schrodinger equation? Variables are separate now!! Divide both sides by ψ(x)φ(t) Temporal part t and x are independent Total wave function Consider only case in which P.E. is time-independent Separation Constant The Temporal Part, φ(t) Solution (see Appendix K) Ae i(kx-ωt) ~ Ae -iωt, ω = C/h

5 Temporal part Total wave function Temporal part Total wave function The probability density is time-independent i.e. the whereabouts of the particle don t change with time in any observable way Oops!! Its time dependence disappears!! Stationary States The probability density is time-independent Quantum Mechanically, electron is not an accelerating charged particles, but rather a stationary cloud Oops!! Its time dependence disappears!! Stationary States The spatial part of!(x,t) Replace C by E, multiply both sides by ψ(x); The time-independent Schrodinger equation: NOTE:!(x) is Real, but!(x,t) is Complex, because "(t)=e -i#t Spatial part Well-behaved wave functions Total Probability of finding the particle = 1 The procedure by which we ensure that the wave funct. gives a unit probability is called Normalization of!(x,t) Smoothness of!(x,t) Another requirement is that a wave function be

6 Normalization of!(x,t) The particle must be somewhere in the universe at any time (the total probability should be = 1) Smoothness of!(x,t) 1. Continuity of!(x,t) 2. Continuity of (d!(x)/dx)! (x) Smoothness of!(x,t)! (x) Smoothness of!(x,t) x x Discontinuity in!(x) Short wave length become zero wave length. i.e. infinite momentum & K.E. (physically unacceptable ) Discontinuity in!(x) # ~ 1! ( k) =! ( x) e 2$! "# "ikx dx Extremely large k (or short %) "> "> Infinite Momentum impossible

7 !( Gaussian Wave Packet #( x/2" ) ik0x x) = Ae e 2 ~! ( k) =? Summary Temporal part " p =! k xˆ 0 Spatial part? Total wave function Find the Spectral Content : $ ~ 1! ( k) =! ( x) e 2%! #$ #ikx dx Normalization of!(x,t) Smoothness of!(x,t) The Schrodinger Equation for Interacting Particles A Particle Interacting With What? Simplification: The Concept of Potential (replaces all individual particle-particle interactions with a single smooth potential) Why? see next page Energy vs. Position for a mass connected to a spring Smooth & Stationary Function

8 Energy vs. Position for a mass connected to a spring Smooth & Stationary Function Independent of the motion of our Particle 3 rd Newton s Law? Not forbidden Bound states is one in which a particle's motion is restricted by an external force to finite region of space In Quantum Mechanics Bound States are Standing Waves In Quantum Mechanics Bound States are Standing Waves

9 The Ground state the lowest energy state is not at E=0 In Quantum Mechanics Bound States are Standing Waves Not forbidden The Ground state - the lowest energy state is not at E=0 Consistent with the Uncertainty Relations: " x" px!! 2 Outline: Today s Lecture Chapter. 5 Bound States: Simple Case Case 1: Particles in a Box The Infinite Well Case 2: The Finite Well Case 3: The Simple Harmonic Oscillator Expectation Values, Uncertainties, and Operators 5.5 Case. I Particle in a Box: The Infinite Well The situation in which the particle-confining U(x) allows the simplest solution of the time-independent Schrödinger equation is called particle in a box, or infinite well CM: simply bounce back & forth QM: standing waves! Schrödinger Eq.

10 Case I: Particle in a box Infinite Potential well Case I: Particle in a box Infinite Potential well E-field in each capacitor exerts a force, F =(-e)e, inward on the electron So its PE is higher outside: U = qv = (-e)(-v 0 ) With total E < ev 0, the electron is bound: Its KE drops to 0 before it can reach a capacitor s outer plate, and it returns in the opposite direction. 1. Continuity of!(x,t) 2. Continuity of (d!(x)/dx)

11 General solution for region I See Appendix K for more detail positive e -αx diverges as x " negative infinite. i.e. mathematically OK, but physically unacceptable. so, D must be zero!!' Region II covers only nagative values of x, where Ce αx is never infinite. x " ±! # "! F = D = 0

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