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Unformatted text preview: Chem 120A Going beyond 1 particle in 1D 03/05 Spring 2007 Lecture 19 READING: Ratner and Schatz, Ch.3 Atkins and Friedman, Ch. 2.12  2.15 So far we have looked at the Particle in a Potential Well in 1Dimension. However, atoms and molecules exist in 3D and in general are composed of multiple particles. So we will now generalize some of our earlier postulates to 2D and 3D and to more than one particle. The commutator relation between the position and momentum operators is now [ ˆ x , ˆ p x ] = i ¯ h [ ˆ y , ˆ p y ] = i ¯ h [ ˆ z , ˆ p z ] = i ¯ h (1) So in a given dimension, position and momentum do not commute. This means that if we measure the position of the paritcle along the xcooridinate and then its momentum in the xdirection, we will not get the same value as if we measure the momentum first in the xdirection and then the position along x. What about position along the xcoordinate and momentum in the ydirection? [ ˆ x , ˆ p y ] = [ ˆ y , ˆ p x ] = (2) So position and momentum operators in different dimensions commute. If we make two measurements but each is along a different dimension, the order of the measurements does not matter. This is a a result of the fact that translations commute (i.e. if I move to x = 6 and then I move to y = 4, I will be in the same spot as If I had moved first to y = 4 and then to x = 6). Our generalized commutator relationship between position and momentum for 1 particle in a multidimensional space is then: [ ˆ r i , ˆ p j ] = i ¯ h δ ij (3) where i and j label the dimension What about measurements on more than one particle? If we measure the position of particle 1 along the xcoordinate, it will not affect the measurement of position along the xcoordinate for particle 2 (as long as the particles are NOT entangled!). Measurements in the same dimension but for different particles commute. Measurements on the same particle but in different dimensions also commute. Only measurements on the same particle and in the same dimension do NOT commmute. Mathematically this is written as: ˆ r α , i , ˆ p β , j = i ¯ h δ ij δ α , β (4) Chem 120A, Spring 2007, Lecture 19 1 where i and j label the dimension and α and β label the particle. We will now write out our position, momentum, and kinetic energy operators in 2D and 3D (using Carte sian coordinates). 2 D 3 D (5) Position ˆ r = x ˆ i + y ˆ j ˆ r = x ˆ i + y ˆ j + z ˆ k (6) Momentum ˆ p = i ¯ h [ ∂ ∂ x ˆ i + ∂ ∂ y ˆ j ] = i ¯ h ∇ ˆ p = i ¯ h [ ∂ ∂ x ˆ i + ∂ ∂ y ˆ j + ∂ ∂ z ˆ k ] = i ¯ h ∇ (7) Kinetic Energy T = ˆ p 2 2 m = ¯ h 2 2 m ∇ 2 T = ˆ p 2 2 m = ¯ h 2 2 m ∇ 2 (8) You have seen that the normalized 1D Particle in a Box wavefunction has units of 1 / √ L (i.e. ψ ( x ) = q 2 L sin ( n π x L ) ). Thus  ψ  2 is a probability density or a probability per unit length (since we are in 1D). The probability of finding the particle at a point along the line between x and x + dx is then...
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This note was uploaded on 02/19/2010 for the course CHEM 120A taught by Professor Whaley during the Spring '07 term at Berkeley.
 Spring '07
 Whaley
 Physical chemistry, Atom, Mole, pH

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