Schrödinger's Equation and Wave Functions in Quantum Mechanics, Slides of Physics

Download Schrödinger's Equation and Wave Functions in Quantum Mechanics and more Slides Physics in PDF only on Docsity! 1 Review of Modern Physics Zeroeth Semester Lecture 15: Docsity.com 2 the wave equation Schrödinger’s Equation linearity and superposition expectation values ―I think that I can safely say that nobody understands quantum mechanics.‖—Richard Feynman (Nobel Prize, 1965) Docsity.com 5 Solutions to the wave equation have the form x y = F t ± . v       The - sign represents waves traveling in the +x direction, and the + sign represents waves traveling in the -x direction. An equivalent functional form is y = F ( k x  t ) . Docsity.com 6 An example of a solution of the wave equation is the wave equivalent to a free particle: (A free particle is one which is not acted on by external forces, including those giving rise to a potential.) j ( kx - ωt )y = A e = A cos ( kx - ωt ) + j A sin ( kx - ωt ) . If we take the real part of y, we have the displacement of waves in a stretched string. (Beiser's equations are slightly different, but equivalent.) Docsity.com 7 What have we accomplished in this section? Not much! We wrote down an ―improved‖ form of the wave equation. We reminded ourselves that if objects are represented by waves, their wave functions must satisfy some form of a wave equation. In this section we introduce Schrödinger's equation, which you can think of as a quantum mechanical statement of the conservation of energy, and which is probably the most important equation of quantum mechanics. 5.3 Schrödinger's Equation: Time-Dependent Form Let's "derive" Schrödinger's equation. ―Where did we get that from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger.‖—Richard Feynman Docsity.com 10 j (kx - ωt) (j/ ) (Px - Et)(x,t) = A e = A e . (j/ ) (Px - Et) (j/ ) (Px - Et)jPA e = A e x         22 2 (j/ ) (Px - Et) (j/ ) (Px - Et) 2 2 jP P A e = A e = - (x,t) . x               Now, ―just for kicks,‖ take 2/x2 The result is -P2/ ħ2 :   Docsity.com 11 Solving* for P2/2m, we get 2 2 2 2 P = - (xt) . x     2 2 2 2 P = - . 2m 2m x     *Why? Because it appears in our ―energy conservation‖ expression. Docsity.com 12 (j/ ) (Px - Et) (j/ ) (Px - Et)-jEA e = A e t             -jE = t     (j/ ) (Px - Et)(x,t) = A e  Once again, ―just for kicks,‖ take /t. The result is –jE / ħ: Remember that Docsity.com 15 The same holds for Schrödinger's equation. It is a postulated first principle, arrived at by observation of physical reality, and believed in because it successfully describes the universe. In other words, we believe it because it works.* *That’s a sign of strength, not weakness. Schrödinger’s equation has been used to explain the previously unexplainable and predict the previously unthought. If you want to replace Schrödinger’s equation with something else, the ―something else‖ must do everything Schrödinger’s equation does, and more. 2 2 2 Ψ j = - + U . t 2m x       Schrödinger's equation is a linear differential equation for the wavefunction . The potential U(x,t), may simply be zero or a numerical constant, or it may be a complicated operator. U(x,t) represents the effects of the universe on the particle. Docsity.com 16 Once we know some boundary conditions for the particle and the potential U(x,t), in principle we can solve for the wave function at any time t and position x. Once we have the wave function, our problem is ―solved.‖ In this chapter we will solve Schrödinger's equation for some simple potentials. Let’s get to work… 2 2 2 Ψ j = - + U . t 2m x       Docsity.com 17 But before that, just to whet your appetite, this is what we are leading up to… from http://www.nearingzero.net Docsity.com 20 Setting LHS = RHS: 2P E = . 2m   2P E = . 2m Duh. We already knew that. Of course! But we needed to check for consistency in the simplest case before spending any more time on Schrödinger's equation. Remember—I told you this is the nonrelativistic version. What can we do that’s useful? Patience! There are still a few ideas to introduce. Docsity.com 21 5.4 Linearity and Superposition 2 2 2 j = - + U t 2m x       In quantum mechanics, the physics and the math seem to be forever entangled. That means we can often gain insight by looking at the math, independent of a particular physical system. Schrödinger's equation is linear in . In other words, it has no terms independent of , and no terms involving higher powers of  or its derivatives. It also means that wave functions ―behave well.‖ Docsity.com 22 As a consequence of this linearity, if 1 and 2 are solutions to Schrödinger's equation, then so is the linear combination 1 1 2 2 = a + a ,   where a1 and a2 are constants. A further consequence of this is that wave functions obey superposition and exhibit interference. (Duh, they represent waves, so they had better do these things, and all other things wavelike.) If a system is represented by a wave function  = a11 + a22, how do we calculate the probability density for ? You can’t just add probabilities! You can’t write P = a1P1 + a2P2, where P1 = 1 * 1 and P2 = 2 *2! Docsity.com 25 Let's begin with an example. Suppose we want to find the average of the set of numbers 1,1,1,2,2,3,3,3,3,4,4,4,4,4. How do you calculate the average? Add the numbers up and divide by the number of numbers? That works, but what if we have zillions of numbers. Is there a way to be clever? The average is        3 2 4 5×1 + ×2 + ×3 + ×4 3 2 4+ + +5 . Sometimes we want to calculate the average value of some measurable quantity. Just as quantum mechanics has its own way of calculating probabilities, quantum mechanics has its own special way of calculating averages. Docsity.com 26 In general, the average of Ni numbers having values xi is i i i i i Nx x = . N   If the variable x is continuous, you replace the sums by integrals. In quantum mechanics, the probability Pi of finding a particle in an interval dx at xi is 2 i iP dx = dx . Think of * as being ―like‖ N (―how much probability‖  ―how many‖). Docsity.com 27 In QM (quantum mechanics) because we are dealing with probabilities, we use the term ―expectation value‖ rather than ―average value.‖ i i i i i Nx x = . N   To get QM average <x>… …replace this by *… …replace this by * with an extra x in it… …and replace these by integrals (because our variables are continuous). Docsity.com 30 In general, the expectation value of any quantity, including operators, is    * - G x = G x dx .     Using the operator expression for momentum also prevents us from using ˆ ˆ* - p = p dx     to claim we’ve found a way to violate the uncertainty principle. the ―hat‖ reminds us momentum is an operator Docsity.com 31 Let's use our wave function as an example. (x) = 3 x for 0 x 1    Refreshing your memory… The expectation value of x is like the average probability of finding a particle at coordinate x. It is the point where you could ―balance‖ the * plot on your fingertips. Docsity.com 32 If I am going to move the pointer along the x-axis then the arrow shown on Figure is depicting the point where the red shaded area would balance on my fingertip. Now let’s see if the math agrees with you.     14 1 1 * 3 - 0 0 0 x 3 x = x dx = 3 x x 3 x dx = 3 x dx = 3 = . 4 4       Docsity.com