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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)

In the following, the deleted and italicized parts are Yuichi's edits.

The Time Independent Schrodinger equation can be separated mapped into a matrix equation for eigenvector and eigen value.

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

This matrix equation can be interpreted in the following way. When a matrix operate on a vector, it will result in a vector. This resulting vector in general is different from the original vector in both direction and the length.

For example, we can think of a simple transformation created by a familiar 2-Dimensional rotation matrix: <math>R = \[\begin{array}{ccc} cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta) \end{array} \]</math>

With this matrix, all vectors are rotated by an angle θ and therefore change their directions. 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. Looking at the eigenvector equation above, it suggests that the two vectors, the original and the one after the transformation by the matrix, M, are in the same direction. i.e. for the matrix M, vector <math>\psi</math> is a special vector which does not change its direction after being operated on by the matrix, M. operator that acts on the wave function and an eigenvector, which equal an eigenvalue multiplied by the same eigenvector:

With this interpretation, the rotation matrix, R, shown above cannot have eigenvectors. This is represented by the fact that this rotation matrix does NOT have real eigenvalues. i.e. there is no vectors whose directions remain the same after the rotation. However, if we expand ourselves and accept complex eigenvalues and vectors with complex components, even this matrix does have two eigen values and corresponding eigen vectors. They are: <math>\cos(\theta)\pm i\sin(\theta)={\rm e}^{\pm i\theta},</math> and <math>\[\begin{array}{c} 1 \\ \pm i \end{array} \] </math>

In order to be able to see or impress this parallel between the time-independent Schrödinger equation and the eigenvector equation, we often write the Schrödinger equation in the following form: Where the Energy is the eigenvalue and the matrix is the Hamiltonian Operator: <math>H\psi = E\psi</math>

It may seem strange to be able to figure out what the unknown wave function AND unknown energy value from a single equation. It almost looks like two unknowns can be figured out from one equation. For the eigenvector equation, the same thing is happening. Unknown eigenvector has n unknowns, in addition to an additional unknown eigenvalue. So altogether there are n+1 unknowns whereas the matrix equation represents n equations. How is this possible? Doesn't it go against the basic of algebra? It turns out that we are not able to determine everything about the eigenvector. We can determine only its direction, but not the length. The length must be determined by other means. For our Schrödinger equation, the same thing happens - the normalization of the wave function is NOT determinable. But our normalization requirement arising from the requirement that the total probability has to be 1 determine the “length” of the wave function.

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.

For most physics applications, the matrix is Hermitian, and consequently, its eigenvectors are perpendicular, so they usually form a orthonormal basis with which all other vectors can be expressed by their linear combinations. A vector x can be resolved decomposed into its perpendicular components projected onto the eigenvectors quite easily. Note that even if the eigenvectors are not orthogonal, as long as they are linearly independent, decomposition of vectors is possible, though figuring out the proper coefficients, c_n, will be trickier.

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.

• more on this topic on tomorrow.

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