Monthly Archives: October 2024

Problem: A smartphone tuning app is able to tune the fifth string of a guitar to \(110\text{ Hz}\) with a precision of \(0.07\text{ Hz}\). Estimate the minimum sampling frequency andsampling time needed for this task. Solution: \[f_s\geq 2\times(110+0.07)\text{ Hz}=220.14\text{ Hz}\] … Continue reading →

Problem #\(1\): Solution #\(1\): Problem: Solution: Problem #\(3\): Solution #\(3\): Problem #\(4\): Solution #\(4\): Problem #\(5\): Solution #\(5\): First, although this tight-binding model looks like a classical model, in fact it arises from the quantum Hamiltonian \(H=E_01-t_1\sum_n(|n+1\rangle\langle n|+|n-1\rangle\langle n|)\) together … Continue reading →

Problem #\(1\): Derive the Maxwell relation for a gas: \[\left(\frac{\partial S}{\partial V}\right)_T=\left(\frac{\partial p}{\partial T}\right)_V\] And explain why Maxwell relations in general should be viewed as much more than just mathematical identities. Solution #\(1\): Here it is clear that one is … Continue reading →

Now suppose that first electron \(e^-\) naturally “burrows” its way down to the ground state \(\textbf n=\textbf k=\textbf 0\) in order to minimize its energy \(E=0\). Now put a second electron \(e^-\) into the box. In reality, the two electrons … Continue reading →

The purpose of this post is to discuss historically one of the first decision problems for which quantum computing was shown to provide an exponential advantage over classical computing. One of the initially striking discrepancies between classical logic gates such … Continue reading →

A qubit is any quantum system with a two-dimensional state space \(\mathcal H\cong\textbf C^2\). In particular because the state space is two-dimensional \(\dim\textbf C^2=2\), the Gram-Schmidt orthogonalization algorithm guarantees the existence of an orthonormal basis \(|0\rangle,|1\rangle\in\mathcal H\) of state vectors … Continue reading →

The purpose of this post is to quickly review some fundamentals of classical computation in order to better appreciate the distinctions between classical computing and quantum computing. Note that the word computation itself, whether classical or quantum, basically just means … Continue reading →

Problem #\(1\): What does the phrase “\(N\) coupled harmonic oscillators” mean? Solution #\(1\): Basically, just think of \(N\) masses \(m_1,m_2,…,m_N\) with some arbitrarily complicated network of springs (each of which could have different spring constants) connecting various pairs of masses … Continue reading →