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lecture06_umn

# lecture06_umn - Lecture 6 Stationary States Most Common...

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Lecture 6 Stationary States Most Common Operators Expectation Values The Uncertainty Principle September 21 2009 Chapter 3 McQuarrie and Simon pages 73-105

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Stationary States Recall that the general solution to the time-dependent Schrödinger equation is the superposition-of-states wave packet defined by (6-1) Let us consider what is required for the probability density at any position in space not to be varying with time . That is, the system is ‘stationary’ or in a ‘stationary state’ . The probability density at a particular position is " x , y , z , t ( ) = c n # n n = 1 \$ % x , y , z ( ) e & iE n t / h
Probability Density at a Particular Position " * x , y , z , t ( ) " x , y , z , t ( ) = c m * # m * m = 1 \$ % x , y , z ( ) e iE m t / h c n # n n = 1 \$ % x , y , z ( ) e & iE n t / h = c m * c n # m * x , y , z ( ) # n x , y , z ( ) e iE m t / h e & iE n t / h m , n = 1 \$ % = c i 2 # i * x , y , z ( ) # i x , y , z ( ) i = 1 \$ % + c m * c n # m * x , y , z ( ) # n x , y , z ( ) e i E m & E n ( ) t / h m n \$ % (6-2) The various terms in the second sum of the bottom equality are not necessarily zero. It is these terms that include a time dependence, so for a system to have a stationary probability density, we must find a way to make every term in the second sum equal to zero .

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Stationary Probability Density There is only one way to do this (unless all states are degenerate): every value of { c } must be zero except for a single one . That is (6-3) Since both the time-dependent and time-independent wave functions are normalized over all space , note that (6-4) Thus, the square modulus of c j must be one (since Ψ is also normalized). " x , y , z , t ( ) = c j # j x , y , z ( ) e \$ iE j t / h " x , y , z , t ( ) " x , y , z , t ( ) = c j # j x , y , z ( ) e \$ iE j t / h c j # j x , y , z ( ) e \$ iE j t / h = c j * e iE j t / h c j e \$ iE j t / h # j x , y , z ( ) # j x , y , z ( ) = c j 2
Stationary States and Time Independent Operators Now, for any operator that does not depend on time , notice that for a stationary state we have (6-5) That is, for a stationary state we can work exclusively with the time- independent spatial wave functions . " x , y , z , t ( ) A " x , y , z , t ( ) = # j x , y , z ( ) e \$ iE j t / h A # j x , y , z ( ) e \$ iE j t / h = e iE j t / h e \$ iE j t / h # j x , y , z ( ) A # j x , y , z ( ) = # j x , y , z ( ) A # j x , y , z ( )

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Table of Observables and Corresponding Operators Observable No. of Dimensions Operator Symbol Operator Position 1 (scalar) x multiply by x 3 (vector) r multiply by r Momentum 1 p (or p x ) " i h i ( ) d dx (always vector) 3 p " i h i # # x + j # # y + k # # z \$ % & ( or " i h ) Kinetic Energy 1 T " h 2 2 m d 2 dx 2 (always scalar) 3 T " h 2 2 m # 2 # x 2 + # 2 # y 2 + # 2 # z 2 \$ % & ( * or " h 2 2 m ) 2 Potential Energy 1 V ( x ) multiply by V ( x ) (always scalar, 3 V ( x , y , z ) multiply by V ( x , y , z ) position dependent) Total Energy, E 1 H T + V (always scalar) 3 H T + V
Table of Observables and Corresponding Operators II Angular Momentum x component (part of 3) L x ! i h y " " z ! z " " y # \$ % & y component (part of 3) L y ! i h z " " x ! x " " z # \$ & z component (part of 3) L z ! i h x " " y ! y " " x # \$ % & scalar square 3 L 2 L x 2 + L y 2 + L z 2 Observable No. of Dimensions Operator Symbol Operator Names, symbols, and operations for the most common operators in both 1 and 3 dimensions.

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Operators and Observables Note that momentum in a single dimension is still a vector quantity , but since there is only one dimension it is often denoted as the scalar ‘component’ times the vector basis for
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