... dispersive.^2.1

This is an important feature of the theory. We know from a wide range of observations that matter moving in a particular medium is not limited to a single velocity.

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... transformations.^2.2

Caution! This is an arbitrary choice. Definitions of Fourier transformations can be found in which the entire $\frac{1}{2\pi}$ is kept either inside or outside of the square brackets. Given the lack of consistency in the literature, it is important to be clear, both as a reader and writer, about which form you are using.

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... function^2.3

For a comprehensive discussion of delta functions, see George Arfken, Mathematical Methods for Physicists, Academic Press (1985).

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... by^2.4

Caution! This approach fails miserably for dispersive waves, because pulses do not retain their forms as they evolve in time.

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... proof,^2.5

See for example, William C. Elmore and Mark A. Heald, Physics of Waves, Dover Publications (1985).

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... literature.^3.1

See for example M. Abramowitz and I. A. Stegun, ed., Handbook of Mathematical Functions, Dover Publications Inc. (1965).

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... approximation^3.2

Wentzel, Kramers, and Brillouin

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... material.^4.1

I follow here the approach of Dudley H. Towne, Wave Phenomena, Dover (1988).

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... has^4.2

The value of $\gamma$ is determined empirically, but it is consistent with the prediction of kinetic theory for an ideal gas consisting of primarily diatomic molecules in a temperature regime in which vibrational states are not excited.

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... respectively.^4.3

This is not the case with a stationary observer, since the wave speed relative to the medium is fixed.

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... observers.^4.4

This asymmetry is not present for electromagnetic waves (the principle of relativity), because there is no medium. This is why the Doppler effect equations for electromagnetic waves differ from those for other classical waves.

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... form^5.1

The real part is implied throughout.

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... by^5.2

Consult your favorite introductory text.

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... by^5.3

Consult your favorite introductory text.

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... is^5.4

Please forgive the use in these notes of $p$ to represent momentum in the context of electromagnetic waves and gauge pressure in the context of acoustic waves.

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....^6.1

For quantum mechanical matter waves, we must take the complex square of $\psi$, and probability takes the place of intensity.

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... source.^6.2

The distance $r$ is a function of $\theta$ if the intensity is measured at different $r$ values for different angles, for example, if the intensity is observed via projection onto a flat screen.

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... function,^6.3

Note that $n = 0$ does not give a root. $\frac{sin(x)}{x} = 1$ at $x = 0$.

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... wavefunction^7.1

The following discussion assumes a one dimensional system. It is worthwhile to consider how one would generalize to three dimensions.

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... average^7.2

Some operators $\hat{O}$ involve differentiation, so the ordering of the integrand is important.

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... time.^7.3

This should be called the ``probability current density.'' It is the analog of the current density encountered in electrodynamics. A properly named ``probability current'' would have units of particles per time.

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... energies^7.4

It is possible for a well ($V_o < 0$) to ``swallow'' some of the $n$ values. That is, we can have $\vert V_o\vert > \frac{\hbar^2 \pi^2}{2m L^2} n^2$ for the first few (or many) $n$ values. Only positive $E_n$ correspond to peaks in transmission.

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... by^7.5

Recall that the direction of the force an electric field exerts on a negative charge is opposite that of the field. Also recall that potential energy increases in the direction opposite that of the force.

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... transformation^7.6

Note that dispersion introduces a second $k^2$ term that was not present in the classical calculation. This adds mathematical complexity and interesting physical effects.

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... well,^7.7

We will postpone indefinitely the issue of normalization, as we are interested in qualitative behavior here.

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... is^7.8

We can replace the spring constant $\kappa$ in the Hooke's law potential energy expression with $m \omega^2$ since

$$\omega = \sqrt{\frac{\kappa}{m}}$$ (7.99)

There is no actual spring in quantum systems, so we express the potential energy in terms of quantities we can measure.

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... other,^7.9

This is a topic for an advanced course. It turns out that this is a property of the any Hamiltonian with a real potential energy function.

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... yields^7.10

Let's agree to remember the time and position dependences of the $c_n$, the $\psi_n$, and $\hat{U}$, and make the notation less explicit but more compact.

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... showed^7.11

See for example Section 9-5 of Tipler and Llewellyn, Modern Physics, 3e, W. H. Freeman and Co. (1999).

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... emission,^7.12

This is a one dimensional translation of Einstein's result for three dimensional systems.

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... proof^7.13

See your favorite advanced undergraduate quantum mechanics text.

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