**Introduction**

Quantum mechanics (QM) is a very peculiar physical theory in that its discovery was not derived from simple laws that are undeniably valid through everyday experience. Unlike classical mechanics, with Newton’s laws of motion describing the actions of macroscopic objects, quantum mechanics would never have taken form without the explanation of blackbody radiation (Planck 1900), the structure of the atom (Bohr 1913), and countless experimental successes (more like the absence of failure). In this lecture, we will focus on the mathematical description of modern quantum mechanics starting with the hypotheses of Max Planck, Albert Einstein, and Louis de Broglie. From these hypotheses, we will derive the equations and concepts that fundamentally describe the microscopic world.

**The Pre-Quantum Wars**

Stephen Hawking described quantum theory as “The dreams that stuff is made of.” Well, such an insubstantial statement is actually not far from truth! In fact, the physical formalism of quantum mechanics was made possible by bold guesses that happened to have triumphed over earlier knowledge (which we now call classical physics). Yet the reason why this “New Physics” claimed victorious is not at all mysterious. You see, physicists prior to Planck, and especially Einstein, were completely sure of themselves that the study of physics was nearly complete. They knew how planets and stars moved, how electromagnetic fields work, and how all other natural sciences followed from just two pillars: MECHANICS and ELECTROMAGNETISM.

However, despite this common arrogance most physicists shared during the late 19th century, there were ongoing feuds over which theories are to die, and which scientists were on the fringe. There was the longest debate of whether light behaved as a wave (Huygens) or particle (Newton), which temporarily ended in favour of waves by the double-slit experiment (Young 1803). In fact, the wave-particle debate was not only confined to light, but ordinary matter as well. This debate resulted in the biggest tragedy in physics –– the death of Ludwig Boltzmann. Boltzmann’s entropic theory of thermodynamics will be remembered as immortal; however, the majority of scientists in Europe at the time rejected his physical intuition of matter being composed of atoms and molecules. Sadly, Boltzmann took his life before every other physicist in Europe submitted to his theory on the existence of atoms. Boltzmann’s picture of the world being composed of discrete particles will pave the way to a deeper understanding of nature for the future of mankind.

**Planck’s Brave New World**

Although Boltzmann’s contributions were immeasurable, the person who truly kickstarted QM would be Planck (even though he was somewhat dubious of his own theory). In 1900, Max Planck described the relationship between the temperature of an object and its emitted frequency using Boltzmann statistics. This was made possible by imposing a postulate unthinkable at the time: objects radiate and absorb in discrete packets of energy (which he called ‘quanta’). In doing so, he gave the relation \[E= h \nu \] where \(E\) is the energy, \(h\) is Planck’s constant, and \(\nu\) is the frequency of light. Although Planck was lucky to have solved the blackbody radiation problem, he did not understand why his “magic postulate” fundamentally worked , nor would he expect to have ignited a revolution in physics.

Five years later, Albert Einstein would make the next step in clarifying Planck’s claim. First, Einstein recognized that Planck’s ‘quantized packet of energy’ as a fundamental unit of energy. By concretizing Planck’s hypothesis, Einstein immortalized the idea by renaming Planck's ‘quanta’ the photon. There are two new features to Einstein’s photon:

1) The photon is a massless particle with energy \(E= h \nu \)

2) Photons travel by the propagation of electromagnetic radiation (light waves)

Thus it is made clear that light is both a particle and a wave.

In 1924, Louis de Broglie furthered the development of QM by making another bold claim: all particles possess wave-particle duality, where the momentum and energy of a particle are given by two relations:

\[p= \hbar k = \frac{h}{\lambda}\] and \[E = \hbar \omega = \frac{{p}^{2}}{2m}\]

\(\hbar = \frac{h}{2 \pi}\)

\(k = \frac{2 \pi}{\lambda}\)

\(\omega = 2 \pi \nu\)

Louis de Broglie's hypothesis radically changed our understanding of matter. His 'matter-waves' generalizes Einstein's photon being both a particle and a wave to every particle we observe in the universe (so fundamentally all particles possess wavelength).

With Planck’s hypothesis of the ‘quanta’, Einstein’s interpretation of the photon, and de Broglie’s wave-particle duality, we are well equipped (from a physical basis) to consider the rest of quantum mechanics. In the next lecture, we will derive the Schrödinger equation out of de Broglie’s relations and the wave-particle duality.

**References**

*Modern Physics for Scientists and Engineers*by John R. Taylor, Chris D. Zafiratos, Michael A. Dubson*Introduction to Quantum Mechanics*by David J. Griffith*Quantum Mechanics Demystified*by David MacMahon*Essential Quantum Mechanics*by Gary E. Bowman*Understanding Physics*by Michael Mansfield, Colm O'Sullivan

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## Comments

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TopNewest@Steven Zheng ,First of all a very thanks.

I wanted to know about quantum mechanics.

I will save your quantum notes ;)

Thanks for so down to earth explanation, that's a rare ability.

Hope to see more notes from you.

kk

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Ancient aliens guy? Have you noticed his hair transformation?

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you mean this?

op

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Like watch this video, don't forget the no.5. Truth or not, I sometimes just enjoy the thrill. But we can discuss the existence of aliens.

h

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There are more ancient paintings and photos.

I sometimes, instead to get the truth, enjoy such stuffs. Remember Pirates Of The Caribbean? Davy Jones and flying Dutchman? They got a whole history out there. Such things have such an impact on me that I also thought of posting problems which will have a touch of fantasy and all.

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Great Note, Sir! @Steven Zheng

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Thank you! Please like and reshare my other notes and problems too!

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Done! Just wanted to ask, are you a Quantum physicist? BTW: I am an aspiring Theoretical Physicist, and you are one of my greatest inspirations! Please advice me for greater magnitude of progress, Thanks! Steven Zheng

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Would you like to share them with me? @Steven Zheng

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click here

Not that many in high school, but the school textbooks. Here are some good reads thoughLog in to reply

@Steven Zheng - thank you for the note, i hope to see more of these and learn, but i have had a doubt since long, when we say particles are waves, do i mean that a particle oscillates paralell to its direction of motion or perpendicularly, that is longitudnal waves or transverse waves, and what made De Broglie think so, it was confirmed by experiments but what exactly was his thought, the common proof is given by combining E=hv and E=mc^2 but then the latter is only for particles and the first one is for light only, so i cant really understand how he joined them. is there an alternative better and rigorious proof?

Thank you

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Believe it or not, the first hypotheses in quantum mechanics were just guesses without proof. For some strange reason, these unprovable statements are always true in experiments. On can build on quantum mechanics using these guesses to develop a fuller picture of our universe, but nothing is inherently proven. Like the theory of evolution, QM is a theory. But this theory is unbelievable accurate and thoroughly backed by evidence. Crazy stuff here.

Actually string theory is said to mathematically prove everything, but it must introduce mathematical theorems unvalidated by any experiment. I guess QM forced humanity to take risks and it worked out fine. String theory is the opposite (from the perspective of the scientific method).

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Hey! A brilliant note! Also, could you direct me to a link or source of learning of topics like Eigenvalues and Eigenfunctions, and their uses in solving Schrodinger's wave equation, if possible?

Thanks! :)

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All matter has wave-particle duality and a frequency that is synonymous with electro-magnetic radiation, ie. matter emits electromagnetic radiation. Electromagnetic radiation has no mass (photons), though perhaps some, but generally it could stand that its mass is located elsewhere, in another dimension. This would make all matter a function of electromagnetic radiation, or a more physical or 3d representation of electromagnetic radiation, hence all matter is light, in degree, but just with more of it jutting into the 3rd dimension, and this seems to support the virtual reality idea of the universe. A holographic reality, based on light, that projects images into solid-looking structures, hence making some things more synonymous with the original light which is causing the holograms. This light is synonymous with space, and with the fabric of the cosmos, or dark and light matter, and the speed of light is the benchmark for movement in the galaxy, the limit that the universe seems to allow, so light is a main component of the galaxy and the highest levels of energy. Matter and energy are synonymous, but light/matter is quantised energy, inside packets for this information. Bending the fabric of the galaxy, using gravity, also bends the light of the galaxy, meaning that all have come from a single source but exist in different forms, or extents of the pure light. Hence, all matter is electromagnetic and does possess and electromagnetic charge. Simple alignment of particles causes electricity to pass through, so light meeting alignment means the transfer of energy.

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@Steven Zheng hey I made a few questions on QM . Could u please check out my questions , Hope you would give me advice about making better questions.

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Steven Zheng could u please make a few more problems. I have learned a bit of QM and a few questions would be awesome.

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Sorry, I was very busy with exams for the past month. Now I am making a study guide for quantum mechanics to sell. Since I have an entire summer to do that, I think I can return to Brilliant for a couple of weeks.

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thank you very much , awesome discussion!!!!

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@Steven Zheng Thanks for the note! I am a bit skeptical about Quantum Mechanics. I mean, even if it were true.. What is it's application? Furthermore, I joined QM in MIT OCW, but was limited due to ignorance of Fourier Analysis. Could you help me?

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Most of our electronics and materials are applications of QM.

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Great note!

But why have you included the image of blackbody radiation curves? As far as I know, I don't think they have anything to do with QM. They are based on Wien's displacement law which was derived in the 1890s (no QM there) and address wavelength.

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Wiens Law led Planck to develop the idea of quantized energy packs.

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