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TheMomentumOperator

# TheMomentumOperator - hx d dx ψ x-i ¯ h d dx xψ x(4 Now...

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The momentum operator Chem 120A Spring 2006 Prof. Harris said in class that the momentum operator in one dimension is given by: ˆ p = - i ¯ h d dx . (1) This expression can be veriFed by considering the commutator for the posi- tion operator, ˆ x , and momentum operator, ˆ p . Recall that these two opera- tors do not communte. The commutator is x, ˆ p ] = ˆ x ˆ p - ˆ p ˆ x = i ¯ h. (2) We can consider the action of the commutator on an arbitrary 1D wavefunc- tion, ψ ( x ): x, ˆ p ] ψ ( x ) = ˆ x ˆ ( x ) - ˆ p ˆ ( x ) (3) Substituting the momentum operator from Equation 1 and x for the position operator: x, ˆ p ] ψ ( x ) = - i ¯
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Unformatted text preview: hx d dx ψ ( x )-(-i ¯ h ) d dx xψ ( x ) . (4) Now we evaluate the derivatives using the product rule: [ˆ x, ˆ p ] ψ ( x ) =-i ¯ hx dψ dx + i ¯ hψ ( x ) + i ¯ hx dψ dx = i ¯ hψ ( x ) (5) ⇒ [ˆ x, ˆ p ] = i ¯ h. (6) By substitution we have shown that the momentum operator for a 1D system as deFned in class obeys the commutation relation and therefore obeys the uncertainty principle for position and momentum. This shows that the momentum operator in Equation 1 is acceptable. 1...
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