What are the qualitative differences between the free particle case and the particle in a box?

The eigenvalues for a free particle are continuous, whereas those for a particle in a box are discrete (dependent on the width of the box). For the particle in the box, the solutions are either "odd" or "even" (the lowest energy solution is even).

How do we find the projection of a function (vector) onto an eigenfunction (element of the basis)?

If there is a function (vector) <math>\Psi</math> and eigenfunction (element of basis) <math>u_n</math> , the projection of <math>\Psi</math> onto <math>u_n</math> is <math>< u_n \mid \Psi > = C_n</math>

How do we expand an arbitrary wavefunction in terms of eigenfunctions?

Assume that the spectrum of <math>\hat A</math> is entirely discrete. If all the eigenvalues <math>a_n</math> of <math>A</math> are non-degenerate there is associated with each of them a unique eigenvector <math>u_n</math>.

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<math>\hat A u_n = a_n u_n </math>

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Since <math>A</math> us Hermitian, the set of <math>u_n</math> is a basis in the wavefunction space. Any wavefunction is expressed with a summation that includes coefficients of expansion and the normalized basis function <math>|u_n></math>. The coefficients of expansion are just as they are in the geometrical analogy. Projections of the function <math>|\Psi></math> on the <math>|u_n></math> direction are given by the inner products between the function <math>|\Psi></math> and the normalized basis function <math>|u_n></math>.

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<math> c_n = <u_n|\Psi> </math>

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<math>\Psi = \sum_n c_n u_n </math>

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How do we predict the probability of obtaining a particular measurement result?

Consider a system whose state is characterized at a given time by the wavefunction <math>\Psi(\vec r, t)</math>. We want to predict the result of a measurement at this time for a physical quantity <math>a</math> assoicated with the observable <math>\hat A</math>. The prediction of a possible outcome will be in terms of probabilities. There can be a prediction of the probability of obtaining a measurement of any eigenvalue of <math>\hat A</math>.

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A fourth postulate states that when the physical quantity <math>a</math> is measured on a system in the normalized state <math>| \Psi (t)></math>, the probability <math>P(a_n)</math> of obtaining the non-degenerate eigenvalue <math>a_n</math> of the corresponding observable is given by an inner product. The function <math>u_n</math> is the normalized eigenvector of <math>A</math> associated with the eigenvalue <math>a_n</math>.

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<math>P(a_n) = |<\Psi|u_n>|^2</math>

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<math>P=<\Psi_n \mid u_n >^2 = C_n^2</math>

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<math>\mbox

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<math> \left \langle \hat A \right \rangle \equiv \left \langle \Psi \mid \hat A \mid \Psi \right \rangle </math>

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How do we find eigenfunctions and eigenvalues of a particle in a box and harmonic oscillator?

Particle in a Box

Quantum Harmonic Oscillator

What is the importance of the harmonic oscillator?

A system exerting a force which is proportional to the displacement <math>F=-kx</math> is called the harmonic oscillator. The force in an harmonic oscillator operates in a direction which is opposite from the displacement. It is called a restoring force.

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<math>-\frac

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<math>-\frac

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The great importance of this particular potential is that any smooth potential in which there is a minimum behaves to some extent like a harmonic oscillator. The Taylor expansion of a smooth potential about the minimum is parabolic in shape.

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Examples of physical systems which can be described by a harmonic oscillator model include vibrational model of diatomic molecules (Leonard Jones) and vibration of atoms in a solid.

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How does a measurement affect the state of a quantum mechanical system?

See the Fifth Postulate.