QMPostulates_000 - Physical Chemistry Course Number: C362 4...

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Unformatted text preview: Physical Chemistry Course Number: C362 4 Postulates of quantum mechanics • Mathematically, we define a quantity, ψ , that completely describes the sys- tem . The quantity, ψ , when represented in terms of coordinates of particles in space, leads to the wavefunction which is represented as ψ ( x ) . The properties of the wavefunction are given below: – The Wavefunction must be continuous. – The wavefunction must have finite values in all space. – The wavefunction must be normalized. That is the integral of the square of the wavefunction over all space must be 1: integraldisplay dxψ ∗ ( x ) ψ ( x ) = 1 (4.1) This condition is extremely important, mathematically. It allows only a certain kind of function to be a wavefunction: ones that are square integrable . – And finally the quantity dxψ ∗ ( x ) ψ ( x ) ≡ dx | ψ ( x ) | 2 is interpreted as the probability density of the system. That is the probability of finding the system in a infinitesimal area of size dx around the point x . • A measurable quantity is described mathematically through an operator, say A . As we will see later, the properties of A are completely defined by what it does to a wavefunction ψ ( x ) . [We will see this in more detail soon.] The associated measured value is given by an integral: ( ψ |A| ψ ) ≡ integraldisplay dxψ ( x ) ∗ A ψ ( x ) (4.2) This is called the expectation value of A . Question: Based on the properties of the wavefunction listed above, how can be rationalize Eq. ( 4.2 )? Later when we solve our first quantum mechanical problem (the particle in a box) we will see how these properties become necessary....
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This note was uploaded on 01/17/2012 for the course C 362 taught by Professor Amarflood during the Winter '11 term at Indiana.

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QMPostulates_000 - Physical Chemistry Course Number: C362 4...

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