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Lect21_HeisUncert

# Lect21_HeisUncert - Quiz 8 For the nth harmonic oscillator...

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Quiz 8 For the n th harmonic oscillator energy eigenstate | n ! , compute the position uncertainty " X. Some useful equations: ( ) 2 A A X + = ! 2 2 Y Y Y ! = " 1 ! = n n n A 1 1 + + = n n n A n = A ( ) n n ! 0 " n ( x ) = # 2 n n ! \$ [ ] % 1/ 2 H n x / \$ ( ) e % x 2 2 \$ 2 ( ) 2 A A i P ! ! = " h

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Lecture 21: Heisenberg Uncertainty Relation PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore
Summary The main results were: 1 ! = n n n A 1 1 + + = n n n A 0 0 = A n = A ( ) n n ! 0 " 0 ( x ) = # \$ [ ] % 1/ 2 e % x 2 2 \$ 2 1 0 = A " 1 ( x ) = 2 # \$ [ ] % 1/ 2 2 x \$ e % x 2 2 \$ 2 " n ( x ) = # 2 n n ! \$ [ ] % 1/ 2 H n x / \$ ( ) e % x 2 2 \$ 2 ) ( x x n X x n ! = ) ( 1 ) ( 2 ) ( 2 1 x n n x x n x n n n ! ! ! ! = " " # " Most efficient way to compute high orbitals numerically: n n n H ) 2 / 1 ( + = ! h Heisenberg Uncertainty Relation Most of us are familiar with the Heisenberg Uncertainty relation between position and momentum: Are the similar relations between other operators? What are the consequences of Uncertainty Relations? 2 h ! " " P X

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Variance The uncertainties are also called ‘variances’ defined as Why is this important? Consider a distribution P (a) , The average of the distribution is:
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