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QMPostulates_000

QMPostulates_000 - Physical Chemistry Course Number C362 4...

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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. Chemistry, Indiana University 18 c circlecopyrt 2011, Srinivasan S. Iyengar (instructor)
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Physical Chemistry Course Number: C362 4.1 Operators
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