Postulates of Quantum Mechanics(from “quantum mechanics” by Claude Cohen-Tannoudji)

6th postulate: The time evolution of the state vector

is governed by the Schroedinger equation

where H(t) is the observable associated with the total energy of the system.

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ψ(t0)

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ihd

dtψ (t) = H(t)ψ (t)

1st postulate: At a fixed time t0, the state of a physical system is defined by specifying a ket

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ψ(t)

Postulates of Quantum Mechanics(from “quantum mechanics” by Claude Cohen-Tannoudji)

2nd postulate: Every measurable physical quantity

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ˆ Q .

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Qis described by an operator This operator is an observable.

3rd postulate: The only possible result of the measurement of a physical quantity is one of the eigenvalues

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Q

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ˆ Q .of the corresponding observable

4th postulate (non-degenerate): When the physical quantity

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Qis measured on a system in the normalized state

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ψ the probability of

obtaining the eigenvalue

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an of the corresponding observable

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ˆ Q is

where is the normalized eigenvector of

associated with the eigenvalue

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an .

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ˆ Q

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un

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P an( ) = un ψ2

Physical interpretation of

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ψ

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ψ 2=ψ *ψ is a probability density. The probability of

finding the particle in the volume element at time is

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dxdydz

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t

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ψ x,y,z, t( )2dxdydz.

General solution for

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ψ x,y,z, t( )

Try separation of variables:

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ψ x, y,z, t( ) =ψ n x,y,z( )θn t( )

Plug into TDSE to arrive at the pair of linked equations:

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θn t( ) = e−iEnt / hand

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ˆ H ψ n = Enψ n

Orthogonality:

For which are different eigenvectors of

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ψa , ψ b

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Hψ n = Enψ n

we have orthogonality:

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ψa*ψ b = 0∫

Let us prove this to introduce the bra/ket notation used in the textbook