Modern Physics: Chapter 8 – Normalizing a Wavefunction

I discussed in the previous chapter you the probability of a particle being detected in the continous range $(x_{0}, x_{1})$ can be calculated from its wavefunction using,

• $\int_{x_{0}}^{x_{1}} \psi^{*} \psi dx$

Now, assuming the particle actually exists, if we essentially look for it at every point in the entire universe we’re bound the detect it somewhere right? Hence the probability of finding the particle in the range $(-\infty, \infty)$ should be $1$. In other words,

• $\int_{-\infty}^{\infty} \psi^{*} \psi dx$ must equal $1$.

So, what if we get a wavefunction, say. $g(x)$ where $\int_{-\infty}^{\infty} g^{*} g dx$ doesn’t equal $1$ and equals some other number $k$ instead? Then we can make the function’s probability equal $1$ bu multiplying it with a constant $A$, such that the corrected wavefunction is:

• $\psi(x) = A \times g(x)$

This process is called normalizing the wavefunction. But how do we figure out what the value of $A$ is? Notice that now that $\psi(x) = A \times g(x)$,

• $\int_{-\infty}^{\infty} \psi^{*} \psi dx$
• $\Rightarrow \int_{-\infty}^{\infty} |A|^2 g^{*}(x) g(x) dx$
• $\Rightarrow |A|^2 \int_{-\infty}^{\infty} g^{*}(x) g(x) dx$

Now we already know that $\int_{-\infty}^{\infty} g^{*} g dx = k$, so the above expression simplifies to:

• $\Rightarrow |A|^2 \int_{-\infty}^{\infty} g^{*}(x) g(x) dx \Rightarrow |A|^2 \times k$

Now to ensure $\int_{-\infty}^{\infty} \psi^{*} \psi dx = 1$, we need a value for $A$ such that

• $|A|^2 \times k = 1$  so
• $|A|^2 = \frac{1}{k}$
• $\Rightarrow |A| = \frac{1}{\sqrt{k}}$

Modern Physics: Chapter 7 – Introduction to Wavefunctions

XKCD/849

So I already mentioned that everything has a matter-wave associated with it. Developing the idea a bit further, we get the concept of a wavefunction which, by convention, is denoted by $\psi$.

Each physical system is described by a state function which determines all can be known about the system. … The wavefunction has to be finite and single valued throughout position space, and furthermore, it must also be a continuous and continuously differentiable function.

Furthermore, a wavefunction is generally a complex function i.e it consists of both a real part and an imaginary part.

• $\psi (x, t)$ – A wavefunction

To get the probably of a particle being at a position $x$, you have to get a real value out of the complex wavefunction. Hence the probably of finding a particle described by a wavefunction $\psi$ is the product of the function $\psi$ at $x$ and its complex conjugate $\psi^{*}$ at $x$.

• $\psi (x_{0}) \psi^{*}(x_{0})$ – probability of finding the particle at $x_{0}$.

To get the probability from a range $x \in (x_{0}, x_{1})$ where $x$ is a continous variable, you can take the integral of $\psi^{*}\psi$ from $x_{0}$ to $x_{1}$:

• $P(x) = \int_{x_{0}}^{x_{1}} \psi^{*}(x) \psi(x) dx$

Modern Physics: Chapter 6 – The Heisenberg Uncertainty Principle

XKCD/824

One of the best explanations I’ve found of the Heisenberg uncertainty principle is in Volume III of the Feynman Lectures on Physics. Read the following sections (which I’ve linked) to understand the general concept behind it:

Essentially, there is an inverse relation between the width of a wave-packet, $\Delta x$ and the range of wavenumbers of the waves you’ll need to superpose to generate that wave-packet, $\Delta k$. In general, the more localized a wavepacket is, the more waves you need to add to create it. Here’s an animation to show what I mean:

We can describe this relation as:

• $\Delta x \Delta k \sim 1$

Now, since

• $p = \hbar k \Rightarrow \Delta p = \hbar \Delta k$

Multiplying the first relation with $\hbar$, we get

• $\Delta x \hbar \Delta k \sim \hbar$
• $\Rightarrow \Delta x \Delta p \sim \hbar$

or more precisely,

• $\Delta x \Delta p \ge \hbar$

Which is the Heisenberg Uncertainty Principle.

Modern Physics: Chapter 5 – Fourier Transform I

XKCD – Fourier

To create a truly localized wavepacket we need to superpose not just two but an infinite amount of sinusoidal waves whose wavelengths and amplitudes vary in a continuous spectrum. To do that, we need to learn about what’s called the Fourier integral:

• $f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} a(k) e^{ikx} dk$

where

• $a(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-ikx} dx$

Why an exponential function instead of sine and cosine? This is because we just used Euler’s formula:

• $e^{ix} = cos(x) + isin(x)$

and it’s derivations:

• $cos(x) = \frac{e^{ix} + e^{-ix}}{2}$
• $sin(x) = \frac{e^{ix} - e^{-ix}}{2i}$

to replace the sines and cosines with the exponential function.

Essentially, this gives us a framework for expressing practically any function as a superposition of harmonic waves (just plug in the function $f(x)$ in the expression for $a(k)$ and then plug in $a(k)$ into the expression for the Fourier integral). This is known as the Fourier transform.

Modern Physics: Chapter 4 – Wave Groups

In the last chapter, I discussed matter waves. Now, when one thinks about waves in general, one visualizes plane waves. Essentially something like this:

Now, the mathematical function for a moving plane wave is of the form

• $f(x, t) = Acos(kx - wt)$

where $k = \frac{2 \pi}{\lambda}$ and $w = 2 \pi f$

You can use the following matlab script to generate a moving plane wave:


%Date: March 27, 2015
%Generating Moving Plane Waves

syms f x t;

A = 3;
k = 2;
w = 2;

f(x, t) = A*cos(k*x - w*t);

ix = 0:0.1:10;

for t = 0:0.1:60
tic;
iy = double(f(ix, t));
plot(ix, iy);
pause(0.1 - toc);
end



Since the intensity matter waves describe the probability of finding a particle at a specific point in space, and since particles are limited to specific regions of space (the probability of finding them can’t just be equal at every point in the universe, the earth for example is far far far far far far more likely to be found in its orbit in the solar system than in the andromeda galaxy), plane waves of infinite extend and the same amplitude throughout are an incorrect representation for a moving particle.

The matter wave associated with a localized moving particle is, in fact, a wave packet. This is what a wave-packet looks like:

Wave packets are created by the super-position of waves with similar but not exactly equal wave-numbers and angular frequencies. In other words, the mathematical function for a simple wave-packet created by the super-position of two waves is something like:

$f(x, t) = Acos(k_{1}x - w_{1}t) + Acos(k_{2}x + w_{2}t)$

You can use the following matlab script to generate a moving wave-packet:

%Author: Muhammad A. Tirmazi
%Date: March 27, 2015
%Generating Moving Wave Groups

syms f g h x t;

A = 3;
k1 = 4;
w1 = 4;
k2 = 4.3;
w2 = 4.3;

f(x, t) = A*cos(k1*x - w1*t);
g(x, t) = A*cos(k2*x - w2*t);
h(x, t) = f(x, t) + g(x, t);

ix = 0:0.1:20;

for t = 0:0.1:60
tic;
iy = double(h(ix, t));
plot(ix, iy);
pause(0.1 - toc);
end


The mathematical function describing a simple wave-packet can be simplified by using the identity

• $cos a = cos b = 2 \times cos \frac{1}{2}(a - b) \times cos \frac{1}{2}(a + b)$

This results in a function of the form:

• $f(x, t) = 2Acos(\frac{\Delta x}{2}k - \frac{\Delta w}{2}t) \times cos(\frac{k_{1} + k_{2}}{2}x - \frac{w_{1} + w_{2}}{2}t)$

where $\Delta k = k_{2} - k_{1}$ and $\Delta w = w_{2} - w_{1}$.

However, we still have a problem with this function. A simple superposition of two slightly different waves generates wave-pulses that repeat at a constant interval. To represent a localized moving particle, we need a localized wave-packet. Essentially, we need something like this:

Modern Physics: Chapter 3 – Making (Non)Sense of the Double Slit Experiment

So let’s get this straight. An electron gives a wave-life diffraction pattern in the double slit experiment. It clearly shows wave behavior. However, as soon as you put detectors in front of the slits and try to detect the electron, its pattern changes to a bullet-like particle pattern. So is an electron a wave or a particle? Well… it’s both and neither.

Wave-Particle Duality

What we have understood till now is that an electron seems to behave like a way till it gets detected. After detection it behaves like a particle. A workable hypotheses might be to consider an electron (and everything else) as both a wave and a particle. A question arises, though. We already know how to calculate the particle properties of the electron such as momentum ( $p = mv$ ), but if it’s a wave, how do we calculate its wavelength?

Matter Waves

The French Physicist, Louis de Broglie postulated that the wavelength of the ‘matter wave’ associated with a particle (such as an electron, proton or even you, me, the earth, Jupiter and chickens) can be calculated using the following relation:

• $\lambda = \frac{h}{p}$

where $\lambda$ is the wavelength of the associated matter wave, $p$ is the momentum and $h$ is what’s called the Planck constant.

Modern Physics: Chapter 2 – Double Slit Experiment II

XKCD – Nerd Sniping

In the last chapter I mentioned that the maxima and minima formed with electrons in the double slit experiment were probability maxima/minima. In fact, the intensity of the wave displayed as the result of the experiment is proportional to the probability of finding the electron at a given position.

The wave, in this case, is a function of the position. Let’s call it $\psi (x)$ . Since $I \propto A^2$ , the probability of finding the electron at a given position $x$ is proportional to $|\psi^2(x)|$. $\psi(x)$ is called a wavefunction.

One Electron at a Time

If you send one electron at a time, it gets detected at one point on the screen. However, if you keep detecting the positions of the individual electrons coming per unit time and record them, you seen find the same pattern emerging.

Here’s a nice animation to show you what I mean: LINK

Detectors in Front of Slits

If you had detectors in front of the slits (even if the detectors are very gentle), the interference pattern disappears. Surprisingly, as soon as it gets detected at the slits, the electron starts giving the expected bullet-like particle pattern on the screen instead of the wave pattern:

Modern Physics: Chapter 1 – Double Slit Experiment I

The results of the double slit experiment challenge the classical concept of the electron being a particle and the concept of nature being divided into particles and waves. Essentially you place a source of whatever you want to conduct the double slit experiment with behind a barrier that only has two narrow slits and then place a detector/screen behind that barrier and record where and when the source gets detected on the screen.

Bullets

If you conduct the experiment with bullets, you should get the following result:

Where $P_{1}(x), P_{2}(x) and P_{12}(x)$ are the probabilities of finding the particle at a given position x when the first slit is open, the second slit is open and both slits are open respectively. In the case of bullets, it is apparent that:

• $P_{12}(x) = P_{1}(x) + P_{2}(x)$

But life isn’t that simple in general. If you conduct the same experiment with waves, you get a somewhat different result.

Waves

If you repeat the pattern with waves, you get the following results:

when $I_{1}(x), I_{2}(x) and I_{12}(x)$ are the intensities of the waves at a given position x when the first slit is open, the second slit is open and both slits are open respectively. Over here,

• $I_{12}(x)$ is clearly not simply equal to $I_{1}(x) + I_{2}(x)$.

This is because the waves interfere with each other either constructively or destructively at different places forming interference maxima and minima at different points.

Electrons

Electrons are clearly particles right? The pattern we get by conducting the experiment with electrons will obviously be the same as the one formed by bullets, right? Wrong! When conducting the experiment with electrons, you get a result more like the one we got with waves than the one we got with bullets.

Interference maxima and minima are formed. Note that unlike the experiment with waves, these are probability maxima/minima, not intensity maxima/minima.

But… electrons were particles, right? Wrong. There’s more to the story than that. To quote Shakespeare,

There are more things in heaven and earth, Horatio,
Than are dreamt of in your philosophy.

~ William Shakespeare – Hamlet

Modern Physics – Introduction

Credits: XKCD

The following is adapted from a lecture by Dr. Sabieh Anwar cited at the end of this post:

Why Study Quantum Physics?

A common question that comes to our mind is: “Why are we studying Modern Physics?” After all, most of us want to become Electrical Engineers because that’s the way the wind is blowing these days. ‘Because that’s where the wind is blowing and we are trying to aimlessly follow the wind.So the question is, in a way, why exactly are non-Physics majors enrolled into this course?’ [translated] The reason for that is that Modern Physics … is one way to look at Nature. Modern Physics has transformed the way we look at Nature. [It] has transformed the way we invent technology. It has transformed our philosophical understanding of our surroundings. So it has transformed our outlook of life. And it’s a very embracing concept … If you look at electronic circuits, almost 75% of the revenue that comes these days in microelectronics is based on devices and inventions that are built on concepts learnt from Quantum Physics …

If we talk about lasers, ‘2010 marked the 50th anniversary of the laser. The first laser was made in 1960. So now we have lasers which are used in DVD-roms and CD players. How do lasers work?’ [translated] What is Silicon? Almost 25% of the Earth’s crust is made of Silicon. Such an abundant material. But why is Silicon so important in the electronic industry? What special features does silicon, or germanium, possess? What is the Big Bang experiment? What are X-Rays? How does diagnostic radiology work? We have a fractured bone. We visit the hospital, and an “X-ray” is performed. How do solar cells work? So all of these are concepts which are vital for an understanding of today’s technology, and an understanding of nature itself. A fundamental understanding of nature.

‘You took courses on Mechanics and Electricity & Magnetism in Freshman year. That was the classical way of looking at Nature.’ [translated] And it’s a very important perspective. With this perspective you can understand a large part of nature. You can understand and decipher how a large number of inventions work. The industrial revolution was based on Mechanics and Heat & Thermodynamics. The entire communications revolution that emerged with the discovery or the invention of radio waves by Heinrich Hertz in the 1890s, that uses concepts of electricity and magnetism. Every one of us carries a cell phone today. You need electricity and magnetism to understand how cell-phones work. Building new cell-phones or optimizing cell-phones. These courses will give you the basics of a large number of devices and concepts in Nature. But, there are certain gaps. The classical picture that was given to you in your first year was incomplete. …. It contains some gaps. There are some things that cannot be explained by it. And now when we’ll study Modern Physics, this will be your first curtain-raiser, your first introduction to non-classical Physics. And classical Physics is a subset of non-classical Physics. If I try to make a diagram:

So what we’ve learnt in the first year is classical Physics. … So now we enter the realm of non-classical Physics which is basically Quantum Mechanics or Quantum Physics. Now this Quantum Physics is a superset of classical Physics. When you look at classical physics, you’re actually looking at average non-classical behavior. So what’s really happening is you have this non-classical realm and you’re averaging this non-classical behavior to observe what is called classical behavior.

– Sabieh Anwar, Lecture 1-A, Modern Physics (2011), School of Science and Engineering, LUMS.

(For oppressed people in idiotic totalitarian countries where youtube is banned (like my country): http://goo.gl/HUkTru )

Surviving SSE – Semester 1

Edit: Also, Happy New Year! 🙂
Tho’ much is taken, much abides; and tho’
We are not now that strength which in old days
Moved earth and heaven, that which we are, we are;
One equal temper of heroic hearts,
Made weak by time and fate, but strong in will
To strive, to seek, to find, and not to yield.
~”Ulysses” by Alfred, Lord Tennyson (Last five verses)
I’m studying at the School of Science and Engineering (SSE) of the Lahore University of Management Sciences (LUMS). I haven’t technically emerged as a survivor yet since I haven’t taken my final exams yet but I’m almost a survivor (assuming I don’t fail the final exams and drop out). LUMS, and university journeys in general, are different for everyone. Some sail through pretty easily, others struggle. Some even abandon the journey half way. So an account of the first semester of my personal journey won’t really be of much help to other people and their own journeys through college life, but one can always try to relate and make analogies.

So I arrived at LUMS in August and attended the first day of the orientation. The speeches by the vice chancellor and the deans were nice. Then began what is called the “O-Week”. Lots of people say the o-week is their best time at LUMS but to be honest I didn’t enjoy it very much. Luckily, if I recall correctly, after the second or third day I realized I wasn’t having much fun and kind of dumped my o-week group. THAT is when the fun started. I explored the LUMS School of Science and Engineering  (SSE)  and emailed Dr. Amer Iqbal, a Theoretical Physicist in the Physics department who I respect and look up to, requesting an appointment. He told me I could come the very next morning to his office in the Physics department. I met him there, discussed the development of Physics and the prospects of becoming a Theoretical Physicist and had a great time. I even took a selfie with him.

I spent most of the rest of the semester roaming around and studying in the Physics department  (I also roamed around the physics labs until I was thrown out twice by one of the lab instructors. She was scary. I still roam around the physics labs on days when she is absent). Ironically, by the end of the semester I realized that, even though I ultimately want to become a Theoretical Physicist, what I really want to major in as an undergraduate is Mathematics. I can always minor in Physics, of course, (and maybe even Computer Science or Economics, which might be helpful in taking Actuarial Examinations) but the field of knowledge I really really want to pursue right now is Mathematics.

Here’s some advice for surviving the first semester:

1. Don’t go to a lot of parties and don’t join a lot of societies. Trust me. I personally didn’t go to any party or join any society all semester but that’s only because I’m a lifeless loser. But keep it minimal. Seriously.
2. Don’t listen to fellow classmates. Take whatever they say with a bit of salt. If your friend tells you to enroll in a course he or she thinks is cool, don’t enroll in it unless you’re actually interested. Taking a course just because your best friend is taking it is the very definition of stupidity.
3. Listen to your faculty-advisors. They give awesome advice because they’re faculty members and have gone through the same struggles and know the system. Whenever you need guidance on something, email them and arrange a meeting. Also help them guide you with deciding your major. Ah, and speaking of majors…
4. Don’t make a biased decision, choose intelligently. When I came to LUMS I was interested in Physics and had a lot of exposure to computer programming.. Yet, I don’t think I’ll blindly choose either Computer Science or Physics as my major. In fact I’m more leaning towards majoring in Mathematics.
5. Don’t give in to peer pressure. There’ll be LOTS and LOTS of people asking you why you’re doing this and shouldn’t you be doing that and you should try this and you’re an idiot for trying that. Don’t listen to them. Make your own decisions.
6. Make friends with the Bossman of SSE. There’s this cool bearded guy with an American accent you might see roaming around SSE. His name is Furqan. He’s the bossman of SSE. Make friends with him. He’s intelligent but also very nice and humble.

One thing I learned after coming here is that to be successful, you have to be humble. You have to be humble enough to consider the possibility that you could be wrong and the other person could be right. You have to be humble enough to objectively consider the evidence and form an unbiased conclusion. That is the key to the scientific method. Realizing that you could be wrong and changing your hypotheses when experimental evidence conflicts with it. In Mathematics especially, being receptive and humble is important. As Paul Dirac said,

If you are receptive and humble, mathematics will lead you by the hand. Again and again, when I have been at a loss how to proceed, I have just had to wait until I have felt the mathematics led me by the hand. It has led me along an unexpected path, a path where new vistas open up, a path leading to new territory, where one can set up a base of operations, from which one can survey the surroundings and plan future progress.

And finally, don’t lose your passion in the journey. Always remember why you’re studying whatever you’re studying. If you do anything, do it with conviction and motivation.