Announcements. The Schrodinger Equation for Interacting Particles. Lecture 16 Chapter. 5 Bound States: Simple Case

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Announcements! HW5: 4-36, 41, 48, 53, 63, 66! Mid-term Exam: 10/16 Ch 1,2,3,4,5(part) *** Course Web Page ** http://highenergy.phys.ttu.

5 Bound States: Simple Case! To make QM useful in real application,!

** interaction of object with its surrounding Outline: Lecture 15 Chapter. 5 Bound States: Simple Case!

A Review of Classical Bound States (First)!Case 1: Particles in a Box The Infinite Well! Case 2: The Finite Well!

1 Announcements! HW5: 4-36, 41, 48, 53, 63, 66! Mid-term Exam: 10/16 Ch 1,2,3,4,5(part) *** Course Web Page ** Lecture Notes, HW Assignments, Schedule for thephysics Colloquium, etc.. Purpose: Lecture 16 Chapter. 5 Bound States: Simple Case! 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. ** interaction of object with its surrounding Outline: Lecture 15 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 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)

express as a product of Standard Math. Technique; Separation of variables!

!! To determine the behavior of particle in (1) CM: solve F = m(d2r/dt2)

Schrödinger eq. for!(x,t), given knowledge of P.E.

, A: allows us to break a differential eq.

2 Adding P.E. Key Assumption: Factorization of the wave function Wave function may be express as a product of Standard Math. Technique; Separation of variables!!time-dependent Schrödinger Eq.!! To determine the behavior of particle in (1) CM: solve F = m(d2r/dt2) for r, given knowledge of Net external F on particle in (2) QM: solve the Schrödinger eq. for!(x,t), given knowledge of P.E., U(x) 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? t and x are independent and factoring out terms constant w.r.t. the partial derivatives Consider only case in which P.E. is time-independent Separation Constant Divide both sides by!(x)"(t) The Temporal Part, "(t) Variables are separate now!! Solution (see Appendix K) Ae i(kx-#t) ~ Ae i#t ; # = C/h

3 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!(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) Another requirement is that a wave function be

Continuity of!(x,t) 2. Continuity of (d!(x)/dx)! (x)! (x) x x Discontinuity in!

4 Normalization of!(x,t) The particle must be somewhere in the universe at any time (the total probability should be = 1) 1. Continuity of!(x,t) 2. Continuity of (d!(x)/dx)! (x)! (x) 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

Summary Temporal part Outline: Today s Lecture Chapter.

Case 2: The Finite Well! Case 3: The Simple Harmonic Oscillator!

I Particle in a Box: The Infinite Well The situation in which the particle-confining U(x) allows the simplest

bounce back & forth QM: standing waves! Schrödinger Eq.

5 Summary Temporal part Outline: Today s Lecture Chapter. 5 Bound States: Simple Case Total wave function Spatial part!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 Normalization of!(x,t) 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. 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.

6 Case I: Particle in a box Infinite Potential well 1. Continuity of!(x,t) 2. Continuity of (d!(x)/dx) General solution for region I See Appendix K for more detail

7 e -$x diverges as x " negative infinite. i.e. mathematically OK, but physically unacceptable. so, D must be zero!!% positive Region II covers only nagative values of x, where Ce $x is never infinite. x " ±! # "! F = D = 0

Chapter. 5 Bound States: Simple Case

Chapter. 5 Bound States: Simple Case Announcement Course webpage http://highenergy.phys.ttu.edu/~slee/2402/ 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