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lecture04_umn - Lecture 4 The Wave Function for a Material...

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Lecture 4 •The Wave Function for a Material System •The Time-independent Schrödinger Equation •What is a Wave Function? •Collapse of the Wave Function September 16 2009 Pages 31-72 McQuarrie and Simon
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Wave Function for a Material System Erwin Schrödinger, in 1926, proposed a way to meld the many quantum observations up to that point with a wavelike description of matter . He started from the classical wave equation (here for a single dimension) (4-1) where Ψ is a wave function which has an amplitude for any specification of x (position) and t (time) , and c is the velocity of the wave (wavelength times frequency). A general solution to this equation is (4-2) where C is an arbitrary multiplicative constant, i is , λ is the wavelength and ν is the frequency. " 2 # x , t ( ) x 2 = 1 c 2 2 # x , t ( ) t 2 " x , t ( ) = Ce 2 # i x $ % & t ( ) * + , 1
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Wave Function Description To verify, note that (4-3) and (4-4) and c = λν . Schrödinger decided to use the de Broglie wavelength for λ (3- 21) and to relate the frequency to the Planck energy(2-4) . " 2 # x , t ( ) x 2 = $ 4 C % 2 & 2 e 2 i x $ t ( ) * + , - 2 # x , t ( ) t 2 = $ 4 C 2 2 e 2 i x $ t ( ) * + , - = h p Ε = h ν (2-4) (3-21)
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Differential equation He looked for a differential equation having the solution (4-5) where p is the momentum and E is the energy . For simplicity, we will take the arbitrary constant C as one. If we consider differentiating eq. 4-5 once with respect to time we have (4-6) or (4-7) "# x , t ( ) " t = $ iE h e i xp $ Et h % & ( ) * " h i #$ x , t ( ) # t = E $ x , t ( ) " x , t ( ) = Ce i xp # Et h $ % & ( )
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Information contained in the Wave Function (4-7) The total energy E at a given time may be expressed as a sum of kinetic and potential energy , both of which depend only on x and not on t : (4-8) where p is the momentum , m is the mass , and V is the potential energy , which may be thought of as an outside influence on the system being described by the wave function Ψ . " h i #$ x , t ( ) # t = p 2 2 m + V x ( ) % & ( ) * $ x , t ( ) " h i x , t ( ) t = E $ x , t ( )
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Information contained in the Wave Function Note that the idea here is to have Ψ be a function that contains information about the system contained within it . That is, we would like things like momentum and energy to be themselves determinable from the wave function . Wave function (4-5) " h i #$ x , t ( ) # t = p 2 2 m + V x ( ) % & ( ) * $ x , t ( ) (4-8) " x , t ( ) = e i xp # Et h $ % & ( )
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How can we determine the momentum from Ψ ? Differentiate once with respect to x: (4-9) Write this in "operator" formalism as (4-10) or (4-11) "# x , t ( ) " x = ip h e i xp $ Et h % & ( ) * x # x , t ( ) = ip h # x , t ( ) h i x = p " x , t ( ) = e i xp # Et h $ % & ( )
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Momentum Operator We say that the operator that can be applied to the wave function in order to determine the momentum is to differentiate once with respect to x and then multiply by h-bar over i . In this case, it is straightforward to show that the square of the momentum (needed for the kinetic energy) can be derived from operating twice with the momentum operator , i.e., (4-12) If we include this result in equation 4-8 we have (4-13) This is the one-dimensional, time-dependent Schrödinger equation .
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lecture04_umn - Lecture 4 The Wave Function for a Material...

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