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Quantum Physics 2005 Notes-4 The Schrodinger Equation (Chapters 6 + 7) Notes 4 Quantum Physics F2005 1 The Schrodinger Equation Hˆ ! ! Eˆ! 2 2 h " "! # ! " V ( x, t )! ! ih 2 "t 2m "x • We can use the relationship between time and space derivatives to explore the behavior of the wavefunctions. Notes 4 Quantum Physics F2005 2 Moving particles and probability flow 2 "P ( x, t ) "! *! "! * " ! h " ! * * ! !! "! !! # 2 "t "t "t "t 2 im " x h " 2! * " ! 2im "x 2 h * " 2! " 2! * " h * "! "! * !# #! !# # ! ! ! 2 2 "x "x "x 2im "x "x 2im h * "! "! * ! ! , we have, Defining, j ( x, t ) ! # 2im "x "x "P ( x, t ) "j ( x, t ) !# "t "x where j ( x, t ) is called the probability current density. Notes 4 Quantum Physics F2005 3 Interpreting probability flow • We can understand this equation if we think about probability in a finite region and current flow into and out of this region. " "b P(# a, b $ , t ) ! ∫ P( x, t )dx ! j (a, t ) # j (b, t ) "t "t a This is a conservation law that states that the only way the probability of finding a particle in a region can change is if the probability of finding it outside the region increases accordingly.. Notes 4 Quantum Physics F2005 4 Probability current for a pure momentum state Taking $ p 0 ( x, t ) ! Aei ( p0 x # E0t ) / h h * "! "! * ! j ( x, t ) ! # ! 2im "x "x 2 hA ! 2im 2 # i ( p0 x # E0t ) / h ip0 i ( p0 x # E0t ) / h #ip0 # i ( p0 x # E0t ) / h i ( p0 x # E0t ) / h e # e e e h h 2 h A 2ip0 A 2 ! ! p0 ! A v0 2im h m probability current is constant in space, therefore what flows into a volume must flow out. Notes 4 Quantum Physics F2005 5 Probability flow in three dimensions We are interested in the probability of finding the particle in a finite volume V. r PV ! ∫∫∫ P(r , t )d% r r #i h * r r r r r * r $ (r , t )&$ (r , t ) # $ ( r , t )&$ ( r , t ) , so We define j (r , t ) ! 2m "P r " &j ! 0 "t r r " r ∫∫∫ V P(r , t )d% " ∫∫ j ' da ! 0 "t A % Notes 4 & Quantum Physics F2005 6 Time evolution of a wave packet -1 • A wave packet is a linear combination of an infinite number of extended singlewavelength with infinitesimally differing wavenumbers. 1 ) i % kx #'t & $ ( x, t ) ! A k e dk ( ) ∫ 2( #) 1 ) # i % kx & dk where A(k ) ! ∫ $ ( x, 0)e 2( #) • This definition allows us to predict the time evolution of the wave packet. Notes 4 Quantum Physics F2005 7 Time evolution of a wave packet – 2 Gaussian example Assuming the spatial part of the wavefunction at t ! 0 is a Normal Gaussian function centered at 0 and moving with average wavenumber k0 ; 1/ 2 1 $ ( x, 0) ! * x 2( e ik0 x # x 2 / 4* x2 e . 1/ 2 1 So A(k ) ! 2( 1 ) i ( k0 # k ) x # x2 / 4* x2 e dx ∫e * x 2( #) (We have already solved for A(k) in Notes 2, page 33. ) 1 4 2 2 #* x2 ( k # k0 )2 A(k ) ! * x e ( Notes 4 Quantum Physics F2005 8 Time evolution of a wave packet – 3 Gaussian example • Now, since we know A(k), we can solve for the time evolution of $(x,t). i 1 ) A k e ( ) $ ( x, t ) ! ∫ 2( #) 2 x 3 kx # k 2 t / 2 m 1 4 ) * x2 #* x2 ( k # k0 ) 2 i dk ! 3 ∫ e e 2( #) kx # k 2t / 2 m dk 1 4 * i k0 x # k02t / 2 m ) 2 h 2 hk 0 ! # " # # e i t u i t ix exp * ∫ x u du 2m #) m 2( which is an integral of a standard form, ∫ exp(#+ u 2 # , u )du. 1 4 * x2 i $ ( x, t ) ! 3 e 2( Notes 4 k0 x # k02 t / 2 m hk 0 x t # # ( m ' exp h h 4 * 2 " i * x2 " i t t x 2m 2m Quantum Physics F2005 9 Time evolution of a wave packet – 4 Gaussian example • An animation of a travelling quantum wave packet can be found at: travelling_gaussian.mws • Notes: • Look at the probability distribution. You can see the motion of the center of the wavepacket is the same for whatever initial width you choose. The speed only depends on the initial choice of k0. • When the initial packet is narrower, the packet spreads more quickly. You can see why this is by looking at the real part of the wave packet (where you can see the waves.) • Short wave components travel faster than longer wave components, as you can see in the display of the real part of psi. • This effect can be observed in the transport of light pulses down fiber optics. Notes 4 Quantum Physics F2005 10 Thus far in our story, • We have shown some properties of wave functions and state functions. – Some rules governing the form of the wave functions and some plausible wave functions. – Given a wave function to describe a particle, how do we deduce observable properties? – What is a plausible wave equation? • We will now start finding solutions to the wave equation under various well-studied conditions. Notes 4 Quantum Physics F2005 11 Things you should understand and be able to do* • Understand and do simple calculations related to the important experiments listed in Notes 1 (Photoelectric, Compton, Diffraction) • Understand and use the Einstein and DeBroglie equations relating particle properties (kinetic energy and momentum) to wave properties (frequency and wavelength) (p. 39 in Morrison). • Understand and perform simple calculations using the Heidenberg Uncertainty Principle (p. 9, Morrison) • Do calculations with probability distributions (normalization, expectation value, variance, and standard deviation). You should also be able to look at a graph of probability and make a reasonable guess of average position and standard deviation. *(on a quiz, for example) Notes 4 Quantum Physics F2005 12 Things you should understand and be able to do –2 • Understand the concept of the complex state function and its relation to probability, especially the wave function and the probability of finding the particle somewhere. • Know the physical limitations on the form of wave functions. (p. 78) • Recognize a pure momentum state (single wavelength, travelling wave) and pick out the wavelength, wave number, frequency, period, and phase velocity. (pp. 107 ff) • Transform back and forth between the wave function (real space) representation and the momentum amplitude function (momentum space) representation of a quantum state. – Set up a transform in each direction. – Perform the math for some simpler cases. Use symmetry to simplify or avoid calculation. – Have a rudimentary understanding of how the real space wave function is related to the amplitude function. • especially, the Heisenberg Uncertainty Principle Notes 4 Quantum Physics F2005 13 Things you should understand and be able to do -3 • Understand the concepts of the wave packet and group velocity. (p. 113) • Be able to perform simple operator arithmatic.(p. 159) • Know and love useful operators for important observables in real space (p. 173) and momentum space (position, momentum, kinetic energy, total energy) • Know the Schrodinger equation and how one simplifies it into the time independent Schrodinger equation. Notes 4 Quantum Physics F2005 14 Another motivational interlude • Many of the really amazing apparent paradoxes of physical understanding have roots in quantum physics. (A particle interfering with itself is odd. A particle being everywhere at once. Quantum cryptography.) • The understanding of many of the most common things is based in quantum physics. (Light emission by a fluorescent bulb. Lasers. Semiconductors. The folding of proteins. Photosynthesis. Chemical processes.) • Developing new nanotechnology requires an understanding of quantum physics. Notes 4 Quantum Physics F2005 15 A special set of solutions – Constant V The full TDSE: "$ ( x, t ) h2 " 2 # $ ( x, t ) " V ( x, t ) $ ( x, t ) ! i h 2 2m "x "t Let's assume that V does not change with time: "$ ( x, t ) h2 " 2 # $ ( x, t ) " V ( x ) $ ( x, t ) ! i h 2 2m "x "t Now we'll search for solutions using the separation of variables method: $ ( x, t ) ! ! ( x)% (t ) Notes 4 Quantum Physics F2005 16 Constant-V solutions "! ( x )% (t ) h 2 " 2 (! ( x)% (t )) # " V ( x)! ( x)% (t ) ! ih 2 2m "x "t " 2! ( x) "% (t ) h2 # " ( ) ( ) ( ) ! ( ) V x x t x i % (t ) ! % ! h 2m "x 2 "t Divide both sides by $ ( x, t ) "% (t ) ! ( x) h 2 % (t ) " 2! ( x) # " V ( x) ! ih 2 2m ! ( x)% (t ) "x "t ! ( x)% (t ) Now we see that the left is only a fn of x and the right only t. The only way this equation can be true for all x and t is if each side is equal to the same constant. Notes 4 Quantum Physics F2005 17 Constant-V solutions So now we have two separated equations: h 2 1 " 2! ( x) # " V ( x) ! + 2 2m ! ( x) "x 1 "% (t ) ih !+ % (t ) "t Looking at the t equation: "% (t ) ih # +% (t ) ! 0 "t yields % (t ) ! % (0)e # i+ t / h and by association with the Einstein-DeBroglie relations: + ! h' ! E Notes 4 Quantum Physics F2005 18 Constant-V solutions So now we have: $ ( x, t ) ! ! ( x)% (t ) ! ! ( x)e # iEt / h Where the total energy is a constant The new equation in x, of the particle. h 2 " 2! ( x) # " V ( x)! ( x) ! E! ( x) 2 2m "x is called the time-independent Schrodinger Equation The wave function solutions to the timeindependent SE are called stationary states because the energy is constant Notes 4 Quantum Physics F2005 19 Special solutions for the onedimensional Schrodinger equation with constant potential: Stationary States Notes 4 Quantum Physics F2005 20 An unconfined particle: Constant potential – V=0 everywhere Notes 4 Quantum Physics F2005 21 Free Space: V= same everywhere Stationary states of the free particle wave -1 " 2! ( x) 2m( E # V ) ! ( x) ! 0 " 2 2 h "x should look familiar from your diff eq course. "2 y 2 or Schaum's Outline 2 " k y ! 0 . "x The solutions are sin kx, cos kx, eikx , or e # ikx 2m( E # V ) where k ! h2 By convention, we set V ! 0 for free space. Notes 4 Quantum Physics F2005 22 The general solution vs the specific case The free particle wave -2 • There are an infinite number of possible solutions to the free space Schrodinger equation. All we have found is the relation between the possible time solutions and the possible space solutions. • We need to give more information about the state for you to limit the set of possible solutions. – If we specify the energy, E, then the set of possible k’s is limited to two possibilities (+ and -), but this still leaves us with sine and cosine, or +k, -k solutions. – We need to be given other limitations, such as the value of the wavefunction at certain points/ certain times. • We will spend a significant portion of this course solving special cases, given interesting V(x) and boundary conditions and/or initial conditions. Notes 4 Quantum Physics F2005 23 Wavefunction for a free particle wave –3 The spatial part of the wave function is a linear combination of a complete set of solutions. Choosing the exponential function representation, ! ( x) ! aeikx " be # ikx so, $ ( x, t ) ! ! ( x)% (t ) ! % aeikx " be# ikx & eiEt / h ! aei ( kx "'t ) " be # i ( kx #'t ) If we specific the direction of propagation to be +x, then $ ( x, t ) ! ! ( x)% (t ) ! be # ikx eiEt / h ! be# i ( kx #'t ) . Looking at the probability density: $ * $ ! b 2 ! constant ) 2 ) Let's try normalizing this: 1= ∫ $ * $dx ! b ∫ dx -) -) This means we have to let b ⇒ 0 as x ⇒ ). Notes 4 Quantum Physics F2005 24 Wavefunction for a free particle wave –4 momentum Since the position is completely indeterminate, let's see what the momentum is: ) ) " * 2 i ( kx #'t ) # i ( kx #'t ) $ $ # i dx b e i ( ik ) e dx h h ∫ ∫ "x p ! #) ! #) ! hk ) * 2 ∫ $ $dx b ∫ dx #) #) 2 ) "2 # i ( kx #'t ) 2 2 " i ( kx #'t ) $ # dx dx b e e h ∫ $ #h ∫ 2 2 2 2 "x "x ! #) ) ! #) k ! h ) 2 * $ $ b dx ∫ ∫ dx ) p2 ) * 2 #) #) So, p 2 # p 2 2 2 2 ! h k # % hk & ! 0 This state has a perfectly defined momentum. Notes 4 Quantum Physics F2005 25 Wavefunction for a free particle – 5 Probability current j ( x, t ) - #ih * " " * $ $ # $ $ 2m "x "x #ihb 2 i ( kx #'t ) e ! (#ik )e # i ( kx #'t ) # e# i ( kx #'t )ike# i ( kx #'t ) 2m 2 hk 2 p !b !b ! Pv m m where P is the probability density Notes 4 Quantum Physics F2005 26 A particle confined to a region of space: The Particle in a Box with Infinitely High Walls Notes 4 Quantum Physics F2005 27 Stationary states of a particle in a box -1 Outside the box, $ ( x, t )=0. Inside the box, V=0, so: $ ( x, t ) ! # a cos(kx) " b sin(kx) $ eiEt / h V=0 V=) x=0 V=) x=L or $ ( x, t ) ! ceikx " de # ikx eiEt / h (we are free to choose which one to use) Boundary conditions: ! (x . 0)=0; Notes 4 ! (x / L)=0 Note that I have chosen to place the boundaries at 0 and L, rather than +/- L/2 as we did previously. I have done this to provide a slightly different approach and solutions, but the basic result is the same Quantum Physics F2005 28 Particle in a box - 2 Starting with the x=0 boundary condition: ! (0) ! a sin 0 " b cos 0 ! 0 The only way this can be true is if b=0. Now for the x=L boundary condition: ! ( L) ! a sin kL ! 0 This can be true if a=0 (no wave function at all) or if kL ! n( . ! n ( x) ! a sin % kn x & and normalizing gives: ! n ( x) ! Notes 4 2 sin % kn x & L Quantum Physics F2005 29 Particle in a box - 3 • The imposition of the two boundary conditions that confine the wavefunction to a region of space leads to quantization of the stationary state particle energies. 2 n( ! n ( x) ! sin x L L Each stationary state has a unique energy: 2 2 h 2 kn2 h ( 2 En ! !n 2m 2mL2 Notes 4 Quantum Physics F2005 30 Particle in a box - 4 • If we chose to place the boundaries of the box at (L/2, we would have found spatial wave functions like those in the book: ! n( e ) ( x) ! ! (o) n ( x) ! 2 n( cos L L x for n=odd 2 n( sin L L x for n=even • The corresponding energy level for a given n is the same as we found for the 0,L box. Notes 4 Quantum Physics F2005 31 Particle in a box – 5 Wave functions for stationary states • Note that however we choose to specify the box, the states have definite even or odd parity about the center of symmetry of the potential. • The existence of this symmetry in the potential led to simple solutions for the wavefunctions. ! ! P P Notes 4 ! ! P P Quantum Physics F2005 32 Particle in a box – 6 Momentum distribution for stationary states • In a previous homework (and in Notes 3), we deduced the momentum amplitude functions for these wavefunctions. n( kL n( kL sin # sin " 1 2 2 2 2 # A(k ) ! for even n n( L( n( # k k " L L P(k) Notes 4 Quantum Physics F2005 33 Particle in a box – 7 Expectation values for stationary states By observing symmetries and from our previous work on particle in a box wave functions, we know: Expectation value of position: xn ! the center of the box 1 1 # 2 2 Uncertainty in position: x ! L 12 2n ( Expectation value of momentum: p ! 0 2 n Uncertainty in momentum: Notes 4 2 n p n( h ! L Quantum Physics F2005 34 Particle in a box –8 Expectation value and uncertainty in energy 2 ˆ p The kinetic energy operator is: Tˆ ! . 2m Since we previously solved for pn2 we can easily solve for T . pn2 2 1 n( h Tn ! ! 2 m 2m L The lowest allowed state has n=1, so expectation value of the kinetic energy is always > zero. Another remarkable observation is that: T 2 # T 2 ! 0. There is no uncertainty in the energy! Notes 4 Quantum Physics F2005 35 Particle in a box – 9 Probability current 2 n( sin L L x e # i'nt #i h * " " * j ( x, t ) $ $ # $ $ 2m "x "x #ih 2 ! # k sin kn x sin knx # k sin kn x sin knx $ ! 0 2m L everywhere in the box. $ n ( x) ! Notes 4 Quantum Physics F2005 36 A problem with this example • A small monster in the closet here is that these wavefunctions do not satisfy all of the physical limitations on wavefunctions that we specified earlier – continuous? – yes – normalizable? – yes – continuous derivative? – no, there is a kink at the boundaries • This situation has arisen because we have actually specified an infinite potential energy discontinuity at the boundaries. Notes 4 Quantum Physics F2005 37 A general observation about energy levels • We have found for the particle in the box that the energy is quantized – it can only take on certain discrete values that are related to the dimensions of the box and the mass of the particle. • We found no such quantization for the free particle plabe wave. • It is a general observation that situations in which particles are bound give rise to quantized energies whereas unbound systems lead to an energy continuum. Notes 4 Quantum Physics F2005 38 Another observation about energy The Time Independent Schrodinger Equation for a free particle also leads to another useful observation: h2 " 2 Inspecting # ! ! E! we see that the left hand side 2 2m "x is a measure of the curvature of the wavefunction, whereas the right hand side is the energy. Basically, the curvier the wave function, the more energy is has. This is useful of you want to get an idea of what the wavefunction might look like for a special potential shape. Notes 4 Quantum Physics F2005 39 An addition to the language: Eigenvalue equations, Eigenfunctions Eigenvalues The Time Independent Schrodinger equation is a second-order, ordinary differential equation in x. It is linear and homogeneous and has 2 independent solutions. Hˆ ! ! E! is known in mathematics as an eigenvalue equation because the operator returns a constant times the original function. ! is called the eigenfunction of the operator Hˆ E is the corresponding eigenvalue for H! Notes 4 Quantum Physics F2005 40 The next set of notes will deal with several specific cases Step Potential Barrier Potential Harmonic Well Potential Particle in a 3D Box (Quantum Dot) Notes 4 Quantum Physics F2005 41 Notes 4 Quantum Physics F2005 42