TheMomentumOperator - hx d dx ( x )-(-i h ) d dx x ( x ) ....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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 ¯
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: hx d dx ( x )-(-i h ) d dx x ( x ) . (4) Now we evaluate the derivatives using the product rule: [ x, p ] ( x ) =-i hx d dx + i h ( x ) + i hx d dx = i h ( x ) (5) [ x, 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...
View Full Document

This note was uploaded on 01/25/2011 for the course CHEM 120A taught by Professor Whaley during the Spring '07 term at University of California, Berkeley.

Ask a homework question - tutors are online