5210-chap2 - Chapter 2 Quantum Theory The Postulates of...

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Chapter 2 Quantum Theory
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The Postulates of Quantum Mechanics Postulate 1: ψ ( x, y, z, t ) is a well-behaved, square integrable function, and | ψ ( x, y , z , t )| 2 dxdydz is the probability of finding the particle between the points ( x, y, z ) and ( x+ dx, y+ dy, z+ dz ) at time t . Postulate 2: For every observable property, there is a linear Hermitian operator that is obtained by expressing the classical form in Cartesian coordinates x and momenta p x and making the replacements: x i p p x x x x = ^ ^ and Postulate 3: The wavefunction ψ ( x, t ) is obtained by solving the equation: Eqn. dinger o Schr dependent - time The ψ ψ .. ^ H t i = Postulate 4: If ψ a is an eigenfunction of the operator  with eigenvalue a , then if we measure the property A for a system whose wave function is ψ a , we always get a as the result. Postulate 5: The average (or expectation) value of an observable A is given by: = τ d d A A ψ * ψ ψ * ψ ^ Where  is the operator associated with the observable. Evidently, if ψ is an eigenfunction of the operator  , then the expectation value is just the eigenvalue.
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( 29 α i Ce t x = , ψ ( 29 Et px - = 1 Definition of ψ and | ψ | 2 At the end of chapter 1 we had shown that a wave-like particle may be described by a wave function, ψ ( x , t ). An example of which might be: The wave function will satisfy the Schrödinger wave equation: 2 2 2 ψ 2 ψ x m t i - = A simple adjustment gives us a 3-dimensional representation: ( 29 ( 29 t z y x t x , , , ψ , ψ ψ 2 ψ 2 ψ 2 2 2 2 2 2 2 2 2 - = + + - = m z y x m t i , , ,
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Definition of ψ and | ψ | 2 The value of the wavefunction ψ ( x,y,z,t ) depends upon the value of the x , y , and z -coordinates of a particle at any given time, t . The uncertainty principle suggests we need a probabilistic description: When we have a continuous distribution we define the probability that the value of x lies in a specified interval x 1 x x 2 : ( 29 = 2 1 ) ( 2 1 x x dx x x x x P ρ where ( x ) is called the probability density distribution. Some Classical Statistics ( x ) P ( x 1 x x 2 ) x 1 x 2
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Definition of ψ and | ψ | 2 – Some More Statistics ( 29 1 ) ( = = - - dx x x P ρ The total probability that - x must be 1, i.e.: If f ( x ) is a function of x we define the expectation value of f , f , as: - = dx x x f f ) ( ) ( We define the mean, x , as: - = dx x x x ) (
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Question: The variable x can have any value in the continuous range 0 ≤ x ≤ 1 with probability density distribution ρ ( x ) = 6 x (1- x ). (i) Derive an expression for the probability that the value of x is not greater than a , P ( x a ). (ii) Confirm that P (0 ≤ x ≤ 1) = 1. (iii) Find the mean x .
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Definition of ψ and | ψ | 2 To obtain a probabilistic description of our wave-like particle we might consider replacing ρ ( x ) with ψ ( x,y,z,t ).
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This note was uploaded on 04/11/2011 for the course CHEM 5210 taught by Professor Staff during the Spring '08 term at North Texas.

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5210-chap2 - Chapter 2 Quantum Theory The Postulates of...

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