Title: ETH_Sem3 / Physik_III / Woche8

Q: A monochromatic wave traveling in free-space has a field that is a solution of the wave equation\[A(x,t) = [...] = [...]\]This wave has some oscillating amplitude [...] in space.
How can we use this to construct a particle that is localized in space?
A: A\cos(kx-\omega t); A\text{ Re}[e^{i(kx-\omega t)}]; everywhere
Back Extra: Use the superposition principle.

Q: What is a wavepacket?
A: How do you call a superposition of different frequency waves?

Q: Total amplitude for a wavepacket of with wavevectors \(k_0 \pm \Delta k\):\[\Psi(x,t) = [...] = [...]\]
What approximation is made and how is it physically motivated?
A: \int_{k_0-\Delta k}^{k_o+\Delta k}Ae^{i(kx-\omega(k)t)}\;dk; 2A\Delta ke^{i(k_0x-\omega_0t)}\text{ sinc}\bigg[\Delta k\bigg(x-\frac{\hbar k_0}{m}t\bigg)\bigg]
Back Extra: Approximation: \(\Delta k\) is small compared to \(k_0\).
This is motivated by the need for the particle to have a somewhat well-defined momentum.

Q: Is real space or k-space better for describing a wave?
A: Neither!
Both are equally good and useful.

Q: How can we switch between real-space functions and k-space functions?
(Also valid for other spaces)
A: With an (inverse) Fourier transform.

The peak of the sinc envelope of \(\Psi(x,t)\) is at\[x_\text{max} = \frac{\hbar k_0}{m}t\]From this we can take\[\frac{dx_\text{max} }{dt} = \frac{\hbar k_0}{m} = \frac{p_0}{m} = v_0\]This is called the group velocity. It is equal to the particle velocity in classical mechanics.

Q: How is the width of the envelope of \(\Psi(x,t)\) defined?
A: As one period of the numerator of the sinc envelope.

Q: Probability of finding the particle between \(x\) and \(dx\) when measured at a time \(t\):
\[P(x,t)dx = [...]\]
A: \vert\Psi(x,t)\vert^2dx
Back Extra: With\[\vert\Psi(x,t)\vert^2 = \Psi^\ast(x,t)\Psi(x,t)\]

Q: What exactly does this equation mean?\[P(x,t)dx = \vert\Psi(x,t)\vert^2dx\]
A: It only makes a statement about the results of physical observations.
It says nothing about what the particle is doing if you don't measure it, or that the particle is in more than one place at the same time.

Q: Why are the (squares of the magnitudes of) wavefunctions normalized?
A: The particle has to have a position / momentum / ... at all time, and the total probability must be 1.

Q: What is Heisenberg's uncertainty principle and what does it mean?
A: \[\Delta p_x\Delta x \geq \frac \hbar2\]Where \(\Delta p_x\) and \(\Delta x\) are the standard deviations of the probability distributions in \(p\) and \(x\).
It means that we cannot simultaneously measure both the position and momentum with arbitrary precision.

Q: What is the purpose of any physical theory, at least according to Prof. Chu?
A: To take the measured properties of a system at one point in time and use it to predict the results of later measurements.

The state of every quantum system can be completely described by a complex valued function called the wavefunction.
It is just a mathematical construction.

Given a set of functions \(\Phi_n(x)\): conditions for a complete orthonormal basis
Orthonormality: \( \int_D\Phi_n^\ast(x)\Phi_m(x)\;dx  =  \delta_{mn} \), (Kronecker delta)
Any function \(\Psi(x)\) defined on the domain \(D\) can be written as: \(\Psi(x) =  \sum_n\Psi_n\Phi_n(x) \)
Where the coefficients can be written as \(\Psi_n = \int_D\Phi_n^\ast(x)\Psi(x)\;dx\)

Q: Condition for completeness of a discrete basis \(\Phi_n(x)\)
\[[...] = [...]\]
A: \sum_n \Phi_n^\ast(x')\Phi_n(x); \delta(x-x')

Properties of delta functions
- \(\delta(x) = 0\) for \(x \neq 0\)
- \(\int_{-a}^a \delta(x)\;dx = 1\) for any \(a > 0\)
- \(\delta(ax) = \frac{1}{\vert a\vert}\delta(x)\) for \(a\in\mathbb{R}\)

Continuous bases
Label the basis functions with a continuous variable \(\alpha:\, \Phi(\alpha,x)\) so that\[\Psi(x) =  \int\tilde\Psi(\alpha)\Phi(\alpha,x)\;d\alpha \]
Orthonormality: \(\int\Phi^\ast(\alpha,x)\Phi(\alpha',x)\;dx  = \delta(\alpha-\alpha')\)
Coefficients: \(\tilde\Psi(\alpha) = \int\Phi^\ast(\alpha,x)\Psi(x)\;dx\)
Completeness: \(\int_{-\infty}^\infty\Phi^\ast(\alpha,x')\Phi(\alpha,x)\,d\alpha = \delta(x-x')\)

Q: What is an observable and how is it represented mathematically?
A: Any observable (with a measurement) property of the system. It is represented as an operator.

Q: What are eigenfunctions, eingenvalues and eigenstates?
A: When an operator applied to a function gives back the same function but multiplied by a constant, we call the function an eigenfunction of the operator, and the constant is called an eigenvalue. The quantum state that the eigenfunction represents is called an eigenstate.

Q: Momentum operator\[\hat p = [...]\]
A: -i\hbar\frac{\partial}{\partial x}

Q: Energy operator\[\hat E = [...]\]
A: i\hbar\frac{\partial}{\partial t}

Q: Position operator\[\hat x = [...]\]
A: x

Each physical observable is represented by a Hermitian operator. The eigenvalues of the operator are the possible results when that observable is measured.

Q: What defines a Hermitian operator?
A: It only has real eigenvalues (when acting on valid wavefunctions)

Q: If a measurement is made of an observable, and the system has [...] of the operator associated with that observable, then the result [...].
A: a wavefunction that is an eigenfunction; will always be the eigenvalue

Q: Spectral theorem
The [...] form [...].
A: eigenfunctions of an operator \(\hat O\) representing an observable; a complete, orthonormal basis

Q: Generalisation:
The probability of measuring a result between \(O\) and \(O+dO\) for an observable \(\hat O\) is given by\[P(O)dO = [...]\]
A: \vert\Psi_O(O,t)\vert^2dO