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Unformatted text preview: Chem 120A Going beyond 1 particle in 1-D 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 1-Dimension. However, atoms and molecules exist in 3-D and in general are composed of multiple particles. So we will now generalize some of our earlier postulates to 2-D and 3-D 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 x-cooridinate and then its momentum in the x-direction, we will not get the same value as if we measure the momentum first in the x-direction and then the position along x. What about position along the x-coordinate and momentum in the y-direction? [ 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 x-coordinate, it will not affect the measurement of position along the x-coordinate 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 2-D and 3-D (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 1-D 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 1-D). The probability of finding the particle at a point along the line between x and x + dx is then...
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