Responsible party: Zeno, Blackbox

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• Reiterating and justification for why <math> \frac{d}{dt} \int f(x)dx=\int \frac{\partial}{\partial t}f(x)dx=0</math> which extends specifically to our application: <math> \frac{d}{dt} \int_{-\infty}^{\infty}\Psi^*\Psi dx=\int_{-\infty}^{\infty}\frac{\partial}{\partial t}(\Psi^*\Psi)dx=0</math>

• Waveforms, Fourier Transform Analysis

CHAPTER 2 Today's Main points:

• Energy Quantization
• Time Independent Schrodinger equation
• Matrices and Linear Algebra: Eigenvalues and Eigenvectors
• Method of Separation of Variables / class of separable solutions
• Simple Harmonic Oscillator - (We'll cover this in another lecture soon)
• Stationary States
• The Free Particle
• Infinite and Finite Square Wells - (We'll go more in depth soon)

• Quantum “particles” exhibit both wavelike and particle-like behavior. They can scatter and interfere like waves, yet can still behave like classical particles in some ways as well.
• Fourier's Theorem states that all continuous functions can be constructed on a specified interval with a series of sine and cosine wave functions (often requiring infinite series). Combinations of a series of waves having certain frequencies and specific scalar weights will conform to any function.
• Quantum particles can be described as “wave packets,” having a frequency and wavelength as well as a position in space, packets of energy with a wave function comprised of a Fourier series of trigonometric functions.

• When a particle is bound by a potential, V, and the total energy, E, is less than V: E < V, there are specific allowed energy levels that a particle can have. A particle can have any of the allowed energies, but not an energy level that is not allowed.
• The Infinite Square Well is an example of a bound state with quantized energy (Griffiths p30-38)
• The Finite Square Well is another bound state with quantized energy (Griffiths p78-82)
• Quantized Energy states do not exist for the ideal Free Particle (Griffiths p59-67) which may have any of a continuous range of energies.

Separable Solutions are a very distinct class of solutions which may be broken down into products of each variable: <math>f(x,t)=g(x)h(t)</math>. Physically these solutions represent a special case and therefore a very small portion of the number of potential solutions that may not be separated into product functions. Mathematically these product solutions can be solved relatively easily with purely analytical theory, the method of Separation of Variables.

The Method of Separation of Variables takes advantage of cases of separable solutions. Derived in Griffiths p24-28, we can separate <math>\Psi(x,t)</math> into two product functions <math>\psi(x)*\phi(t)</math>. With a product solution, we can rearrange and substitute so the Schrodinger equation reads <math>i\hbar\frac{1}{\phi}\frac{d\phi}{dt}=-\frac{\hbar^2}{2m}\frac{1}{\psi}\frac{d^2\psi}{dx^2}+V</math> The key here is that the left side depends only on t and the right side depends only on x. You could vary either t or x and fix the other, and the equation must still be satisfied. This can only be true if both sides are equal to a constant, and furthermore the same constant.

1. Keep in mind that this only works for separable solutions. That is, solutions of the Schrodinger Equation that can be separated in to a product of two functions, each of which only depends on one variable. This is a narrow class of solutions, and potentially very few of all of the solutions that exist would satisfy these conditions so it shouldn't surprise you that this analysis seems to be valid only for a very special case.

If each side of the above separated Schrodinger equation is equal to a constant, E, we can write the time-dependent equation as: <math>\frac{d\phi}{dt}=-\frac{iE}{\hbar}\phi </math> which has the easily obtained exponential solution: <math> \phi(t)=e^{-iEt/\hbar} </math>

The right side is also equal to a constant and is only a function of x, and multiplying through by <math>\phi(x)</math> yields the Time Independent Schrodinger Equation. The key idea in the Method of Separation of Variables is that we've effectively turned a partial differential equation into two ordinary differential equations which we can solve analytically.

As described above and worked out in further detail in Griffiths p25, the Time Independent form is: <math>-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2} + V\psi = E \psi </math> The key features of the Time-Independent form are:

• Every expectation value is constant in time
• The probability density is constant in time (although the wave function does depend on t-see p26)
• These “Stationary States” are states with a Definite Total Energy. The total energy does not change with time.
• The general solution is a linear combination of separable solutions. Each <math>\psi_n(x)</math> has a corresponding exponential <math> \phi(t) </math> with an energy <math> E_n </math>, and a proportionality constant <math> c_n </math> (Griffiths p26-28)

The Time Independent Schrodinger equation can be separated into a matrix operator that acts on the wave function and an eigenvector, which equal an eigenvalue multiplied by the same eigenvector:

<math>[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V] \psi = E \psi </math> ⇒ <math> M \psi = \lambda \psi</math>

Where the Energy is the eigenvalue and the matrix is the Hamiltonian Operator: <math>H\psi = E\psi</math>

We know from Linear Algebra that an n dimensional matrix M and the Eigenvalue/vector equation can be solved for <math>(n-1)</math> variables and <math>\lambda</math>. Multiplication by the matrix M represents a linear transformation of <math>\psi</math>, and the eigenvalue equation represents a transformation that maps all values of <math>\psi</math> to zero.

A simple transformation is the 2-Dimensional rotation matrix: <math>\[\begin{array}{ccc} cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta) \end{array} \]</math>

• For the Eigenvalue equation, the matrix M does not change the direction of the eigenvector <math>\psi</math>, only its magnitude; it is equivalent to multiplication by a constant.

For most physics applications eigenvectors are perpendicular, so a vector x can be resolved into its perpendicular components projected onto the eigenvectors quite easily.

The Hydrogen Atom has an infinite number of Energy levels, so an infinite number of eigenvalues are possible. This also implies that the transformation matrix M can be infinite-dimensional.

• Stationary States have the property that All Expectation Values are Constant in Time. They represent a very special case when the Energy levels are the same and the time dependence cancels upon calculating expectation values.

• The wave function itself can depend on time, but the probability density and expectation values do not because the complex conjugates cancel each other for the same energy.

• Every measurement of Energy will return the Exact same value, E.

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