Physics 256: Lecture Schrodinger Equation. The Effective Wavelength Time Independent Schrodinger Equation
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1 Physics 56: Lecture Schroinger Equation The Effective Wavelength Time Inepenent Schroinger Equation Examples
2 Clicker Question 0: What is true about a particle in the first energy eigenfunction for the quanton in a box? P b a 1 x x A. 1 an only B. 1,, an 3 only C. an 3 only D. 1 an 4 only E. All of the above. 1. If we measure the position we know for sure what we will get. If we measure the energy we know for sure what we will get 3. P tells us the probability of fining the particle between a an b 4. P tells us the probability of fining the particle with Energy E 1
3 Clicker Question 1:
4 Generalizing the e Broglie Relation We want a formalize way to etermine energy eigenfunctions for a 1-D system (given V(x h h p mk As the kinetic energy increases the wavelength ecreases K E V x As the potential increases the kinetic energy ecreases Stuent: I'm having some trouble unerstaning how we efine the wavelength as a function of x
5 TISE We will spen much of the rest of the semester learning to solve for Energy Eigenfunctions given a particular V(x. Now let s see what we can gain by qualitatively stuying this equation m E ( x E V x x E x x E E ( E (
6 Clicker Question : Which coul be solutions of this DE (C is a positive real constant? f ( x A. B. f ( x f ( x A cos Ae C x C x Cf ( x x C. f ( x Ae i C x D. (B an (C E. (A an (C
7 Differential Equations Translation: We will spen much of the rest of the semester solving ifferential equations f (1 ( x ( x Cf ( x f ( x Asin C x f ( x Cf ( x x f ( x B Wave-like functions Ae Cx Be cos Cx Exponential-like functions C x
8 Time-Inepenent Schroinger Equation Concavity ivie by the function gives a measure of the curvature of the function Units = 1/length f ( x / x 1 f (x (x ( (x is efine as the local wavelength (wavelength at a point ( x x C f f C = -4 ( x / x Stuent: How o we solve C in lama square function? Stuent: I am not convince that we can just say that n erivative/function in Q1. can be relate to wavelength just because it has units of length involve.
9 Time-Inepenent Schroinger Equation (TISE h mk h m K h 1 E V x m x 1 4 ( x E ( x / x E h ( x / x E E V m x 4 E ( x
10 h ( x / x E E V m x 4 E ( x
11 Stuent: What oes the EEF physically represent? How oes its behavior relates to the energy the particle has? m ( x E E x E ( x
12 The Schroinger Equation We can re-write the TISE: ( x m E E V x x x E ( Secon erivative is linke to the value of eigenfunction at x an the quantity an the ifference E V(x. The secon erivative at a given x expresses the curvature
13 The Schroinger Equation 1 If E > V(x E is wave-like (curves towars x-axis Classically ll allowe f ( x Cf ( x x If E < V(x E is exponential-like (curves away from x-axis Classically forbien f ( x Cf ( x x
14 The Schroinger Equation We can re-write the TISE: ( x m E E V x x x E ( An energy eigenfunction is always smooth an continuous Smooth = first erivative is continuous
15 More About the Energy Eigenfunctions The local amplitue of the wave-like part of the solution ecreases as the value of E - V(x ( increases Smaller E - V(x ( means a smaller K Classically, a smaller kinetic energy woul imply the particle woul spen more time in that region Quantum mechanically this is manifeste in a larger amplitue Stuent: Why woul the local amplitue ecrease as K increases?
16 Sketching Energy Eigenfunctions 1 Wave-like when classically allowe, exponential-like like when classically forbien Curvature of solutions increases as E - V(x 1 Classically allowe: Larger E - V(x, shorter local (x Classically forbien: Larger E - V(x, ( shorter exp. tail 3 The local amplitue of the wave-like part of the solution ecreases as the value of E - V(x ( increases 4 Solutions are continuous an smooth if V(x is finite 5 Solutions approach zero as x 6 Energy eigenfunctions for boun quantons have an integer number of bumps in the classically allowe region (more bumps = more energy
17 Clicker Question 3: This eigenfunction is: A. correctly rawn incorrectly rawn because Is there anything else wrong? B. it curves incorrectly C. exponential tails are incorrect D. amplitue of wavy part is wrong E. wavelength of wavy part is wrong
18 Clicker Question 4: This eigenfunction is: A. correctly rawn incorrectly rawn because Is there anything else wrong? B. it curves incorrectly C. wavy part oesn t have right number of bumps D. amplitue of wavy part is wrong E. wavelength of wavy part is wrong
19 Why Boun States are Quantize If the energy is off by a little bit the wave function will not be normalizeable i.e not physical, goes off to infinity Therefore, the E is well efine an isolate The energy of a boun state is quantize This is the quantum in quantum mechanics
20
21 Curving away from x-axis
22 Why oes the solution of the Schroinger equation not go to positive/negative infinity for small negative x but big positive x when the energy is not "right"? Curving away from x-axis
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