
Hello World
Hello world! I am Faheel Hashmi and I wish to talk about quantum mechanics. This is a theory that describes the nature at its most basic level, and the nature at this level is very different from how we experience it in our everyday life. I have taught quantum mechanics a couple of times, and… Continue reading

From ket vectors to wavefunction 2
In our previous post Position Operator: From Ket vectors to wavefunctions we have seen that wavefunctions are the scalar product of ket vectors with position eigenkets. More precisely we call these position space wavefunctions. We can also have momentum space wavefunctions which are the scalar product of the ket vectors with momentum operator eigenkets. We… Continue reading

Exercise Problem
A particle in a box of length $latex L&s=2$ is in the state $latex \left\psi\right\rangle&s=2$ such that its wavefunction is $latex \psi(x)=\left\langle x\psi\right\rangle= \sqrt{\frac{2}{L}}\sin\left(\frac{\pi x}{L} \right)&s=2$ What is the probability that the particle is in right half of the box? Solution The probability density associated with the given wave function is $latex \mathrm{p}(x) = \psi(x)^2=… Continue reading

Position Operator: From Ket vectors to wavefunctions
Suppose we are interested in measuring the position of a quantum particle. The observable that will do the job is the position operator. The eigenvalue equation for position operator is $latex \hat x \leftx\right\rangle=x \leftx\right\rangle&s=2$ Here $latex \leftx\right\rangle &s=2$ are position eigenkets and $latex x &s=2$ are the corresponding eigenvalues. This operator is different from… Continue reading

Particle in a box 2
In A Particle in a Box 1 we have seen that we can write the quantum state of a particle in a box in the eigenbasis of energy operator. From now on we will call this the Hamiltonian operator. For a recap the eigenvalue equation for the Hamiltonian for a particle in a box system… Continue reading

Summary of what we have learned
Here is a short summary of what we have learned. Quantum mechanics is a theory that allows us to calculate the probabilities of fundamental interactions taking place in the microscopic world of atoms and molecules. To calculate these probabilities, and to use quantum mechanics we consider the state of the system which is a vector… Continue reading

A Particle in a Box 1
In the previous example we have seen that quantum mechanics allows us to calculate the experimentally verifiable probabilities. In this post we will further build our theory by taking another simple example, and by considering the quantum mechanical description of that system. Let’s consider a particle trapped in a box. Its classical description comprises the position… Continue reading

Nature speaks linear algebra
In our previous post we considered a simple interferometric experiment and claimed that quantum mechanics can account for the experimentally observed detection probabilities in the interferometer. In this post we will see how it is done. The idea was to to attach complex amplitudes to all paths available to the photon. Mathematically it can be… Continue reading

What is quantum mechanics?
A photon approached a semitransparent mirror and asked “Will you let me pass through?”. The mirror kept silent for a minute and then replied “I don’t know. The god has not yet rolled the dice”. Quantum mechanics is a theory that allows us to calculate the probabilities of random events happening in the nature at… Continue reading