Lect07 - Schrdingers Equation and the Particle in a Box (x)...

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Schrödinger’s Equation and the Particle in a Box U= ψ (x) 0 L U= n=1 n=2 x n=3 Midterm Exam this week will cover topics through Lecture 6, and will include Discussion, Homework and Lab topics through HW 3. Important new material will be covered in Discussion Section this week. Be sure to attend.
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Overview Overview z Particle in a “Box” -- matter waves in an infinite square well z Wavefunction normalization z General properties of bound-state wavefunctions
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Last lecture: The time-independent SEQ (in 1D) KE term PE term Total E term ) ( ) ( ) ( ) ( 2 2 2 2 x E x x U dx x d m ψ = + = z Notice that if U(x) = constant , this equation has the simple form: ) x ( C dx d 2 2 ψ = ψ where is a constant that might be positive or negative. ) E U ( 2m C 2 = = For positive C , what is the form of the solution? a) sin kx b) cos kx c) e ax d) e -ax For negative C , what is the form of the solution? a) sin kx b) cos kx c) e ax d) e -ax
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Last lecture: The time-independent SEQ (in 1D) KE term PE term Total E term ) ( ) ( ) ( ) ( 2 2 2 2 x E x x U dx x d m ψ = + = z Notice that if U(x) = constant , this equation has the simple form: ) x ( C dx d 2 2 ψ = ψ where is a constant that might be positive or negative. ) E U ( 2m C 2 = = For positive C , what is the form of the solution? a) sin kx b) cos kx c) e ax d) e -ax For negative C , what is the form of the solution? a) sin kx b) cos kx c) e ax d) e -ax
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Constraints on the form of Constraints on the form of ψ ψ (x) (x) z (x) | 2 corresponds to a physically meaningful quantity – the probability of finding the particle near x. d ψ /dx is related to the momentum probability density. Physically meaningful states must have: ψ (x) must be single-valued, and finite (finite to avoid infinite probability density) Ψ (x ) must be continuous, with finite d ψ /dx (because d ψ /dx is related to the momentum density) In regions with finite potential, d ψ /dx must be continuous, with finite d 2 ψ /dx 2 , to avoid infinite energies. There is usually no significance to the overall sign of ψ (x). (it goes away when we take the absolute square) {In fact, we will see that ψ (x,t) is usually complex!}
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Act 1 Act 1 1. Which of the following hypothetical wavefunctions for a particle in some realistic potential U(x) is acceptable? ψ( x ) x (c) ψ( x ) x (a) ψ( x ) x (b) 2. Which of the following wavefunctions corresponds to a particle more likely to be found on the left side? (c) (b) (a) ψ (x) 0 x ψ (x) 0 x ψ (x) 0 x
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Solution Solution 1. Which of the following hypothetical wavefunctions for a particle in some realistic potential U(x) is acceptable? ψ( x ) x (c) ψ( x ) x (a) ψ( x ) x (b) (b) Acceptable Both ψ( x ) and d ψ/ dx are continuous everywhere (a) Not acceptable ψ( x ) is not continuous at x=0. dx d ψ not defined. d ψ/ dx is not continuous at x=0 (c) Not acceptable
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Solution Solution (a) (b) 2. Which of the following wavefunctions corresponds to a particle more likely to be found on the left side?
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Lect07 - Schrdingers Equation and the Particle in a Box (x)...

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