LectureQ11

LectureQ11 - Chapter 11 Using the form of the Schrdinger...

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4/27/2004 H133 Spring 2004 1 Chapter 11 Using the form of the Schr ö dinger Equation we can learn some qualitative properties of energy eigenfunctions. General Goals for this chapter ± ± ± ± Let’s start with the shapes of the energy eigenfunctions. First recall the Schr ö dinger Equation and let’s write it in the following form If the second derivative (left-hand side) is negative, the function is concave down . If the second derivative is positive the function is concave up. Classically Allowed Region (E – V(x)>0) ± If Ψ E (x) > 0 , ± If Ψ E (x)<0, ± The wave function in the classically allow region always ________________________ [] ) ( ) ( 2 ) ( 2 2 2 x x V E m dx x d E E ψ = ±
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4/27/2004 H133 Spring 2004 2 Shape of Ψ E Classically Forbidden Region: (E – V < 0) ± If Ψ E (x) > 0 , ± If Ψ E (x)<0, ± The wave function in the classically forbidden region always _____________________ These features lead to the general statements that we observed before ± In the classically allowed region the eigenfunction is _____________ ± In the classically forbidden region the eigenfunction is ______________ In addition, from the form of the Schr ö dinger Equation we can also see that: ± Classical Allowed Region: ² The larger E – V , the larger the curvature which implies a shorter wavelength (higher momentum) ± Classically Forbidden Region: ² The large E – V, the shorter (steeper) exponential tails…less probing into forbidden region.
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4/27/2004 H133 Spring 2004 3 Simple Case Consider the very simple case of a square well with finite sides: In the allowed region, the simple sine function satisfies the Schr ö dinger Equation. Let’s see how: Now, this is a solution if the term in the brackets is zero so Notice that we got an expression for b, but not for E or φ . ± We know this Ψ E works but we need to bring some other information into the problem to determine E and φ . Let’s consider the solution in the classically forbidden region. V=V o V=0
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4/27/2004 H133 Spring 2004 4 Square Well In the classically forbidden region we believe the solution is exponential-like. For starts let’s choose an exponential and see if that works. We see that we satisfy the Schr ö dinger Equation if the following condition is meet: Again we determined the parameter β but we still have no more insight into E, C, or B (or φ ). How do we get these? ± “Boundary” conditions: We know that at the walls of the well where the wave function makes the transition from the sine-wave solution to the exponential solution that the wave-function must be smooth and continuous.
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This note was uploaded on 06/03/2011 for the course H 133 taught by Professor Furnstahl during the Spring '11 term at Ohio State.

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LectureQ11 - Chapter 11 Using the form of the Schrdinger...

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