title

Some light quantum mechanics (with minutephysics)

description

The math of superposition and quantum states.
Minutephysics channel: https://www.youtube.com/user/minutephysics
Help fund future projects: https://www.patreon.com/3blue1brown
This video was sponsored by Brilliant: https://brilliant.org/3b1b
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: http://3b1b.co/light-quantum-thanks
Huge thanks to my friend Evan Miyazono, both for encouraging me to do this project, and for helping me understand many things along the way.
This is a simple primer for how the math of quantum mechanics, specifically in the context of polarized light, relates to the math of classical waves, specifically classical electromagnetic waves.
I will say, if you *do* want to go off and learn the math of quantum mechanics, you just can never have too much linear algebra, so check out the series I did at https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab
Mistakes: As several astute commenters have pointed out, the force arrow is pointing the wrong way at 2:18. Thanks for the catch!
*Note on conventions: Throughout this video, I use a single-headed right arrow to represent the horizontal direction. The standard in quantum mechanics is actually to use double-headed arrows for describing polarization states, while single-headed arrows are typically reserved for the context of spin.
What's the difference? Well, using a double-headed arrow to represent the horizontal direction emphasizes that in a quantum mechanical context, there's no distinction between left and right. They each have the same measurable state: horizontal (e.g. they pass through horizontally oriented filters). Once you're in QM, these kets are typically vectors in a more abstract space where vectors are not necessarily spatial directions but instead represent any kind of state.
Because of how I chose to motivate things with classical waves, where it makes sense for this arrow to represent a unit vector in the right direction, rather than the more abstract idea of a horizontal state vector, I chose to stick with the single-headed notation throughout, though this runs slightly against convention.
Music by Vincent Rubinetti:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
------------------
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detail

{'title': 'Some light quantum mechanics (with minutephysics)', 'heatmap': [{'end': 1100.878, 'start': 1087.202, 'weight': 1}], 'summary': "Delves into light and wave mechanics in quantum mechanics, covering topics such as electromagnetic fields, maxwell's equations, light polarization, superposition, and polarized filters, offering insights into the relationship between energy in classical waves and probabilities governing quantum behavior.", 'chapters': [{'end': 106.25, 'segs': [{'end': 106.25, 'src': 'embed', 'start': 57.852, 'weight': 0, 'content': [{'end': 65.834, 'text': "And what we'd like to build to here is the relationship between the energy in a purely classical wave and the probabilities that govern quantum behaviour.", 'start': 57.852, 'duration': 7.982}, {'end': 71.44, 'text': "In fact, we'll actually spend most of the time talking through the pre-quantum understanding of light,", 'start': 66.678, 'duration': 4.762}, {'end': 73.882, 'text': 'since that sets up a lot of the relevant wave mechanics.', 'start': 71.44, 'duration': 2.442}, {'end': 80.805, 'text': 'The thing is, a lot of ideas from quantum mechanics, like describing states as superpositions with various amplitudes and phases,', 'start': 74.382, 'duration': 6.423}, {'end': 86.668, 'text': "come up in the context of classical waves in a way that doesn't involve any of the quantum weirdness people might be familiar with.", 'start': 80.805, 'duration': 5.863}, {'end': 93.973, 'text': "This also helps to appreciate what's actually different in quantum mechanics, namely certain restrictions on how much energy these waves can have,", 'start': 87.168, 'duration': 6.805}, {'end': 98.496, 'text': "how they behave when measured and quantum entanglement, though we won't cover entanglement in this video.", 'start': 93.973, 'duration': 4.523}, {'end': 104.381, 'text': "So we'll start with the late 1800s understanding of light as waves in the electromagnetic field.", 'start': 99.177, 'duration': 5.204}, {'end': 106.25, 'text': "Here, let's break that down a bit.", 'start': 105.029, 'duration': 1.221}], 'summary': 'Exploring the relationship between classical wave energy and quantum behavior, focusing on pre-quantum light understanding and its relevance to wave mechanics.', 'duration': 48.398, 'max_score': 57.852, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b457852.jpg'}], 'start': 2.881, 'title': 'Understanding light and wave mechanics in quantum mechanics', 'summary': 'Discusses the relationship between energy in classical waves and probabilities governing quantum behavior, focusing on the late 1800s understanding of light as waves in the electromagnetic field.', 'chapters': [{'end': 106.25, 'start': 2.881, 'title': 'Quantum mechanics: understanding light and wave mechanics', 'summary': 'Discusses the importance of understanding light and wave mechanics in quantum mechanics, emphasizing the relationship between energy in classical waves and the probabilities governing quantum behavior, with a focus on the late 1800s understanding of light as waves in the electromagnetic field.', 'duration': 103.369, 'highlights': ['The chapter emphasizes the importance of understanding light and wave mechanics in quantum mechanics, particularly the relationship between energy in classical waves and the probabilities governing quantum behavior.', 'The video aims to present a topic that sets the right intuitions for anyone interested in learning quantum mechanics, focusing on the late 1800s understanding of light as waves in the electromagnetic field.', 'The discussion explores the pre-quantum understanding of light, which sets up a lot of the relevant wave mechanics and helps appreciate the differences in quantum mechanics.', 'The chapter mentions the relevance of classical waves in understanding quantum mechanics, highlighting the similarities in describing states as superpositions with various amplitudes and phases.', 'The chapter also briefly touches upon certain restrictions on the energy of waves, their behavior when measured, and quantum entanglement, without covering the latter in the video.']}], 'duration': 103.369, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b42881.jpg', 'highlights': ['The chapter emphasizes the importance of understanding light and wave mechanics in quantum mechanics, particularly the relationship between energy in classical waves and the probabilities governing quantum behavior.', 'The video aims to present a topic that sets the right intuitions for anyone interested in learning quantum mechanics, focusing on the late 1800s understanding of light as waves in the electromagnetic field.', 'The discussion explores the pre-quantum understanding of light, which sets up a lot of the relevant wave mechanics and helps appreciate the differences in quantum mechanics.', 'The chapter mentions the relevance of classical waves in understanding quantum mechanics, highlighting the similarities in describing states as superpositions with various amplitudes and phases.', 'The chapter also briefly touches upon certain restrictions on the energy of waves, their behavior when measured, and quantum entanglement, without covering the latter in the video.']}, {'end': 270.207, 'segs': [{'end': 137.097, 'src': 'embed', 'start': 106.53, 'weight': 1, 'content': [{'end': 108.752, 'text': 'The electric field is a vector field,', 'start': 106.53, 'duration': 2.222}, {'end': 114.456, 'text': 'and that means every point in space has some arrow attached to it indicating the direction and strength of the field.', 'start': 108.752, 'duration': 5.704}, {'end': 119.621, 'text': 'Now, the physical meaning of those arrows is that if you have some charged particle in space,', 'start': 115.057, 'duration': 4.564}, {'end': 123.784, 'text': "there's going to be a force on that particle in the direction of the arrow,", 'start': 119.621, 'duration': 4.163}, {'end': 127.367, 'text': "and it's proportional to the length of the arrow and the specific charge of the particle.", 'start': 123.784, 'duration': 3.583}, {'end': 130.991, 'text': 'Likewise, the magnetic field is another vector field,', 'start': 128.187, 'duration': 2.804}, {'end': 137.097, 'text': 'where now the physical meaning of each arrow is that when a charged particle is moving through that space,', 'start': 130.991, 'duration': 6.106}], 'summary': 'Electric field: vector field with force on charged particle, proportional to arrow length and charge. magnetic field: affects moving charged particle', 'duration': 30.567, 'max_score': 106.53, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4106530.jpg'}, {'end': 189.235, 'src': 'embed', 'start': 160.256, 'weight': 0, 'content': [{'end': 164.939, 'text': "understanding of how these two fields work are Maxwell's equations, which, among other things,", 'start': 160.256, 'duration': 4.683}, {'end': 167.921, 'text': 'describe how each of these fields can cause a change to the other.', 'start': 164.939, 'duration': 2.982}, {'end': 175.606, 'text': "Specifically, what Maxwell's equations tell us is that when the electric field arrows seem to be forming a loop around some region,", 'start': 168.641, 'duration': 6.965}, {'end': 180.949, 'text': 'the magnetic field will be increasing inside that region perpendicular to the plane of the loop.', 'start': 175.606, 'duration': 5.343}, {'end': 189.235, 'text': 'And symmetrically, such a loop in the magnetic field corresponds to a change in the electric field within it perpendicular to the plane of the loop.', 'start': 181.55, 'duration': 7.685}], 'summary': "Maxwell's equations describe the interaction between electric and magnetic fields, showing how changes in one field cause changes in the other.", 'duration': 28.979, 'max_score': 160.256, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4160256.jpg'}, {'end': 240.024, 'src': 'embed', 'start': 215.293, 'weight': 2, 'content': [{'end': 221.199, 'text': 'When you hear the term electromagnetic radiation, which refers to things like radio waves and visible light,', 'start': 215.293, 'duration': 5.906}, {'end': 225.884, 'text': "this is what it's talking about propagating waves in both the electric and magnetic fields.", 'start': 221.199, 'duration': 4.685}, {'end': 231.457, 'text': "Of course, it's now almost mainstream to know of light as electromagnetic radiation,", 'start': 227.014, 'duration': 4.443}, {'end': 234.82, 'text': "but it's neat to think about just how surprising this was in Maxwell's time,", 'start': 231.457, 'duration': 3.363}, {'end': 240.024, 'text': 'that these fields that have to do with forces on charged particles and magnets not only have something to do with light,', 'start': 234.82, 'duration': 5.204}], 'summary': 'Electromagnetic radiation encompasses radio waves and visible light, propagating through electric and magnetic fields.', 'duration': 24.731, 'max_score': 215.293, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4215293.jpg'}], 'start': 106.53, 'title': "Electromagnetic fields and maxwell's equations", 'summary': 'Delves into the physical interpretation of electric and magnetic field vectors, their interaction, and their link to electromagnetic radiation, uncovering the surprising connection between light and these fields.', 'chapters': [{'end': 270.207, 'start': 106.53, 'title': "Electromagnetic fields and maxwell's equations", 'summary': 'Explains the physical meaning of electric and magnetic field vectors, how they interact, and their connection to electromagnetic radiation, including the surprising relationship between light and these fields.', 'duration': 163.677, 'highlights': ["Maxwell's equations describe how changes in electric fields cause changes in magnetic fields, and vice versa, resulting in propagating waves of oscillating electric and magnetic fields.", 'The physical meaning of the arrows in the electric and magnetic fields represents the force on charged particles and the interaction with moving charges, quantitatively related to the specific charge, velocity, and length of the field arrows.', 'The surprising connection between electromagnetic fields and light, where propagating waves in both fields result in the phenomenon of electromagnetic radiation, including radio waves and visible light.']}], 'duration': 163.677, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4106530.jpg', 'highlights': ["Maxwell's equations describe the interaction between electric and magnetic fields, resulting in propagating waves.", 'The arrows in the fields represent force on charged particles and interaction with moving charges.', 'The connection between electromagnetic fields and light leads to electromagnetic radiation.']}, {'end': 434.831, 'segs': [{'end': 298.581, 'src': 'embed', 'start': 270.867, 'weight': 1, 'content': [{'end': 275.59, 'text': "And let's go ahead and ignore the magnetic field right now, just looking at how the electric field oscillates.", 'start': 270.867, 'duration': 4.723}, {'end': 282.933, 'text': "And also, we're only going to focus on one of these vectors oscillating in the plane of the screen, which we'll think of as the XY plane.", 'start': 276.11, 'duration': 6.823}, {'end': 288.456, 'text': 'If it oscillates horizontally, like this, we say that the light is horizontally polarized.', 'start': 283.553, 'duration': 4.903}, {'end': 298.581, 'text': 'So the y-component of this electric field is zero at all times, and we might write the x-component as something like cos,', 'start': 289.036, 'duration': 9.545}], 'summary': 'Analyzing electric field oscillations in xy plane for light polarization.', 'duration': 27.714, 'max_score': 270.867, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4270867.jpg'}, {'end': 364.419, 'src': 'embed', 'start': 335.185, 'weight': 2, 'content': [{'end': 343.651, 'text': "Now by default, cosine only oscillates between negative 1 and 1, so let's put another term in front, a, that gives us the amplitude of this wave.", 'start': 335.185, 'duration': 8.466}, {'end': 348.709, 'text': 'One more thing, just to make things look a little more like they often do in quantum mechanics.', 'start': 344.747, 'duration': 3.962}, {'end': 355.273, 'text': "instead of writing it as a column vector like this, I'm going to separate it out into two different components, using these symbols called kets.", 'start': 348.709, 'duration': 6.564}, {'end': 364.419, 'text': 'This ket here indicates a unit vector in the horizontal direction, and this ket over here represents a unit vector in the vertical direction.', 'start': 355.974, 'duration': 8.445}], 'summary': "Introducing 'a' as amplitude, and using kets for vector representation.", 'duration': 29.234, 'max_score': 335.185, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4335185.jpg'}, {'end': 427.369, 'src': 'embed', 'start': 395.135, 'weight': 0, 'content': [{'end': 399.999, 'text': 'That is, at each point in time, add these two vectors tip to tail to get a new vector.', 'start': 395.135, 'duration': 4.864}, {'end': 406.285, 'text': "Doing this at all points in space and all points in time gives a new, valid solution to Maxwell's equations.", 'start': 400.58, 'duration': 5.705}, {'end': 408.247, 'text': 'At least, this is all true in a vacuum.', 'start': 406.766, 'duration': 1.481}, {'end': 413.006, 'text': "This is because Maxwell's equations in a vacuum are what's called linear equations.", 'start': 408.925, 'duration': 4.081}, {'end': 418.407, 'text': "They're essentially a combination of derivatives acting on the electric and magnetic fields to give zero.", 'start': 413.406, 'duration': 5.001}, {'end': 427.369, 'text': 'So if one field f1 satisfies this equation and another field f2 satisfies it, then their sum f1 plus f2, also satisfies it,', 'start': 418.827, 'duration': 8.542}], 'summary': "Adding vectors tip to tail gives valid solution to maxwell's equations in a vacuum.", 'duration': 32.234, 'max_score': 395.135, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4395135.jpg'}], 'start': 270.867, 'title': "Maxwell's equations and light polarization", 'summary': "Explores the oscillation of the electric field in light polarization, encompassing horizontal and vertical polarization, frequency, amplitude, phase shift, and the combination of distinct waves as valid solutions to maxwell's equations in a vacuum.", 'chapters': [{'end': 434.831, 'start': 270.867, 'title': "Maxwell's equations and light polarization", 'summary': "Discusses the oscillation of the electric field in light polarization, including the concept of horizontal and vertical polarization, frequency, amplitude, phase shift, and the combination of distinct waves as valid solutions to maxwell's equations in a vacuum.", 'duration': 163.964, 'highlights': ['The concept of horizontal and vertical polarization is discussed, including the equations representing each type of polarization.', "The addition of two distinct waves as a valid solution to Maxwell's equations in a vacuum is explained, based on the linearity of the equations.", 'The impact of frequency on the oscillation cycle of the cosine function is described, with lower frequencies leading to longer cycles.']}], 'duration': 163.964, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4270867.jpg', 'highlights': ["The addition of two distinct waves as a valid solution to Maxwell's equations in a vacuum is explained, based on the linearity of the equations.", 'The concept of horizontal and vertical polarization is discussed, including the equations representing each type of polarization.', 'The impact of frequency on the oscillation cycle of the cosine function is described, with lower frequencies leading to longer cycles.']}, {'end': 944.568, 'segs': [{'end': 485.795, 'src': 'embed', 'start': 459.49, 'weight': 4, 'content': [{'end': 467.236, 'text': 'But if the horizontal and vertical components were out of phase with each other, which might happen if you increase the phase shift in one of them,', 'start': 459.49, 'duration': 7.746}, {'end': 469.658, 'text': 'their sum might instead trace out some sort of ellipse.', 'start': 467.236, 'duration': 2.422}, {'end': 476.844, 'text': 'And in the case where the phases are exactly 90 degrees out of sync with each other and the amplitudes are both equal.', 'start': 470.359, 'duration': 6.485}, {'end': 479.365, 'text': 'this is what we call circularly polarized light.', 'start': 476.844, 'duration': 2.521}, {'end': 485.795, 'text': "This, by the way, is why it's important to keep track not just of the amplitude in each direction, but also of the phase.", 'start': 480.226, 'duration': 5.569}], 'summary': 'Adjusting phase shifts can result in circularly polarized light.', 'duration': 26.305, 'max_score': 459.49, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4459490.jpg'}, {'end': 554.623, 'src': 'embed', 'start': 522.092, 'weight': 3, 'content': [{'end': 525.196, 'text': 'at least when both are in phase with each other and they have the same magnitude.', 'start': 522.092, 'duration': 3.104}, {'end': 532.443, 'text': 'Now, the choice of which directions you write things in terms of is called a basis, and which basis is nicest to work with.', 'start': 526.297, 'duration': 6.146}, {'end': 535.486, 'text': "well, that typically depends on what you're actually doing with the light.", 'start': 532.443, 'duration': 3.043}, {'end': 540.27, 'text': 'For example, if you have a polarizing filter like that from a set of polarized sunglasses,', 'start': 535.866, 'duration': 4.404}, {'end': 547.137, 'text': 'The way these work is by absorbing the energy from electromagnetic oscillations in some particular direction.', 'start': 540.851, 'duration': 6.286}, {'end': 554.623, 'text': 'A vertically oriented polarizer, for example, would absorb all of the energy from these waves along the horizontal directions.', 'start': 547.837, 'duration': 6.786}], 'summary': 'Light polarization can be described using a basis, depending on the direction and magnitude, for practical applications.', 'duration': 32.531, 'max_score': 522.092, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4522092.jpg'}, {'end': 642.422, 'src': 'embed', 'start': 618.334, 'weight': 2, 'content': [{'end': 625.482, 'text': 'One way to think of this is that complex numbers are just a very convenient and natural mathematical way to encode an amplitude and a phase with a single value.', 'start': 618.334, 'duration': 7.148}, {'end': 630.528, 'text': "That can make things a little confusing, because it's hard to visualize a pair of complex numbers,", 'start': 626.303, 'duration': 4.225}, {'end': 633.191, 'text': 'which is what would describe a superposition of two base states.', 'start': 630.528, 'duration': 2.663}, {'end': 642.422, 'text': 'But you can think about the use of complex numbers throughout quantum mechanics as a result of its underlying wavy nature and its need to encapsulate the amplitude and the phase for each direction.', 'start': 633.752, 'duration': 8.67}], 'summary': 'Complex numbers encode amplitude and phase in quantum mechanics.', 'duration': 24.088, 'max_score': 618.334, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4618334.jpg'}, {'end': 735.133, 'src': 'embed', 'start': 706.366, 'weight': 0, 'content': [{'end': 713.932, 'text': 'Specifically, the energy of one of these electromagnetic waves always seems to come as an integer multiple of a specific constant times the frequency of that wave.', 'start': 706.366, 'duration': 7.566}, {'end': 719.138, 'text': "We now call this constant Planck's constant, commonly denoting it with the letter h.", 'start': 714.292, 'duration': 4.846}, {'end': 723.884, 'text': 'Physically. what this means is that whenever this wave trades its energy with something else like an electron,', 'start': 719.138, 'duration': 4.746}, {'end': 728.571, 'text': 'the amount of energy it trades off is always an integer multiple of h times its frequency.', 'start': 723.884, 'duration': 4.687}, {'end': 735.133, 'text': 'Importantly, this means there is some minimal, non-zero energy level for waves of a given frequency, hf.', 'start': 729.289, 'duration': 5.844}], 'summary': "Planck's constant h indicates integer multiple energy levels for electromagnetic waves.", 'duration': 28.767, 'max_score': 706.366, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4706366.jpg'}, {'end': 826.368, 'src': 'embed', 'start': 796.115, 'weight': 1, 'content': [{'end': 801.439, 'text': 'And with a different choice in bases, that same state might be described as a superposition of two other directions.', 'start': 796.115, 'duration': 5.324}, {'end': 805.723, 'text': 'All of this is stuff that you would see if you started reading more into quantum mechanics.', 'start': 802.04, 'duration': 3.683}, {'end': 811.247, 'text': 'But this superposition has a different interpretation than before, and it has to.', 'start': 806.303, 'duration': 4.944}, {'end': 815.89, 'text': "Let's say you were thinking of this diagonally polarized photon kind of classically,", 'start': 811.887, 'duration': 4.003}, {'end': 819.473, 'text': 'and you said it has an amplitude of one unit for some appropriate unit system.', 'start': 815.89, 'duration': 3.583}, {'end': 826.368, 'text': 'Well that would make the hypothetical amplitudes of its horizontal and vertical components each the square root of one half.', 'start': 820.165, 'duration': 6.203}], 'summary': 'In quantum mechanics, superposition involves hypothetical amplitudes, like square root of one half for horizontal and vertical components.', 'duration': 30.253, 'max_score': 796.115, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4796115.jpg'}], 'start': 436.792, 'title': 'Superposition and wave nature', 'summary': 'Explains wave superposition, polarization, and the importance of phase in wave description, as well as the concept of basis and its relevance in describing light. it also discusses the concept of superposition in quantum mechanics, the significance of complex numbers, the energy quantization of electromagnetic waves, and the experimental implications of quantum superposition.', 'chapters': [{'end': 579.906, 'start': 436.792, 'title': 'Wave superposition and polarization', 'summary': 'Explains wave superposition, polarization, and the importance of phase in wave description, as well as the concept of basis and its relevance in describing light, with a focus on polarizing filters.', 'duration': 143.114, 'highlights': ['Wave superposition is a sum of two waves, and the resulting superposition can form an ellipse if the horizontal and vertical components are out of phase, and circularly polarized light is produced when the phases are 90 degrees out of sync with equal amplitudes.', 'The phase of the waves affects the way they add together, which is also an important concept in quantum physics.', 'Describing waves using different directions is called a basis, and the choice of basis depends on the specific application, such as analyzing light passing through polarizing filters.']}, {'end': 944.568, 'start': 579.906, 'title': 'Quantum superposition and wave nature', 'summary': 'Discusses the concept of superposition in quantum mechanics, the significance of complex numbers, the energy quantization of electromagnetic waves, and the experimental implications of quantum superposition, including the collapse of superposition upon measurement.', 'duration': 364.662, 'highlights': ["The energy of electromagnetic waves seems to come in discrete amounts, with the energy quantization being a result of Planck's constant, denoted by the letter h, and waves trading energy in integer multiples of h times their frequency.", 'Complex numbers are used to encode the amplitude and phase of quantum states, providing a convenient mathematical representation for superposition.', 'The concept of superposition in quantum mechanics implies that the components of a state, described using complex numbers, have a different interpretation compared to classical waves, and the energy of a wave comes in discrete, indivisible chunks due to quantum mechanics.', 'Experimental implications of quantum superposition include the collapse of a superposition upon measurement, where the superposition collapses to be entirely in one state or another, revealing the probabilistic nature of quantum behavior.']}], 'duration': 507.776, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4436792.jpg', 'highlights': ["The energy of electromagnetic waves is quantized in integer multiples of Planck's constant", 'Experimental implications of quantum superposition reveal the probabilistic nature of quantum behavior', 'Complex numbers encode the amplitude and phase of quantum states, providing a convenient mathematical representation for superposition', 'Describing waves using different directions is called a basis, relevant in analyzing light passing through polarizing filters', 'Wave superposition can form an ellipse or circularly polarized light based on the phase of the waves']}, {'end': 1318.748, 'segs': [{'end': 990.803, 'src': 'embed', 'start': 964.955, 'weight': 0, 'content': [{'end': 969.457, 'text': 'so they actually have a 0% chance of passing a filter oriented horizontally.', 'start': 964.955, 'duration': 4.502}, {'end': 977.917, 'text': 'But if you insert a third filter oriented at a 45 degree angle between the two, it actually lets more light through.', 'start': 970.253, 'duration': 7.664}, {'end': 985.48, 'text': "And what's going on here is that 50% of the photons passing that vertical filter will also pass through the diagonal filter.", 'start': 977.937, 'duration': 7.543}, {'end': 990.803, 'text': "And once they do, they're going to be changed to have a purely diagonal polarization.", 'start': 986.021, 'duration': 4.782}], 'summary': 'Inserting a third filter at a 45-degree angle increases light transmission by 50%.', 'duration': 25.848, 'max_score': 964.955, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4964955.jpg'}, {'end': 1051.711, 'src': 'embed', 'start': 1027.832, 'weight': 3, 'content': [{'end': 1039.16, 'text': 'one specific value we focus on there is the probability that a photon whose polarization is 22.5 degrees off the direction of a filter is going to end up passing through that filter.', 'start': 1027.832, 'duration': 11.328}, {'end': 1043.184, 'text': "Again, it's helpful to think of this wave as having an amplitude of 1,", 'start': 1040.021, 'duration': 3.163}, {'end': 1051.711, 'text': "and then you'd think of the horizontal component as having amplitude sine of 22.5 degrees, which is around 0.38..", 'start': 1043.184, 'duration': 8.527}], 'summary': 'Analyzing the probability of a photon passing through a filter at 22.5 degrees polarization angle.', 'duration': 23.879, 'max_score': 1027.832, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b41027832.jpg'}, {'end': 1108.708, 'src': 'heatmap', 'start': 1078.12, 'weight': 2, 'content': [{'end': 1083.661, 'text': 'And, like we said before, classically this would mean if you pass it through a vertical filter,', 'start': 1078.12, 'duration': 5.541}, {'end': 1086.542, 'text': '15% of its energy is absorbed in the horizontal direction.', 'start': 1083.661, 'duration': 2.881}, {'end': 1092.408, 'text': 'But because the energy of light comes in these discrete quanta that cannot be subdivided.', 'start': 1087.202, 'duration': 5.206}, {'end': 1100.878, 'text': 'instead, what you observe is that 85% of the time the photon passes through entirely and 15% of the time it gets completely blocked.', 'start': 1092.408, 'duration': 8.47}, {'end': 1108.708, 'text': "Now I want to emphasize that the wave equations don't change.", 'start': 1105.745, 'duration': 2.963}], 'summary': 'Light passing through a vertical filter: 85% passes through, 15% gets blocked.', 'duration': 30.588, 'max_score': 1078.12, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b41078120.jpg'}, {'end': 1230.973, 'src': 'embed', 'start': 1203.985, 'weight': 4, 'content': [{'end': 1210.031, 'text': 'Each component in this superposition is given with an amplitude and a phase often encoded as a single, complex number,', 'start': 1203.985, 'duration': 6.046}, {'end': 1213.374, 'text': 'and the need for this phase arises from the wave nature of these objects.', 'start': 1210.031, 'duration': 3.343}, {'end': 1215.496, 'text': 'As with the photon example,', 'start': 1214.094, 'duration': 1.402}, {'end': 1225.104, 'text': 'the choice of how to measure these objects can determine a set of base states where the probability of measuring a particle to be in one of these base states is proportional to the squares of the amplitudes of these numbers.', 'start': 1215.496, 'duration': 9.608}, {'end': 1230.973, 'text': "It's funny to think, though, that if the wavy nature of electrons and other particles was discovered first,", 'start': 1225.805, 'duration': 5.168}], 'summary': 'Superposition components have amplitudes/ phases, affecting measurement probabilities.', 'duration': 26.988, 'max_score': 1203.985, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b41203985.jpg'}], 'start': 944.568, 'title': 'Polarized filters and quantum polarization', 'summary': 'Discusses an experiment showing the impact of filter orientation on light transmission and explores the quantum nature of light, highlighting wave-particle duality and photon polarization probabilities.', 'chapters': [{'end': 1027.832, 'start': 944.568, 'title': 'Polarized filters and light experiment', 'summary': 'Discusses an experiment using polarized sunglasses to demonstrate how the orientation of filters affects the transmission of light, revealing that introducing a third filter at a 45 degree angle between two perpendicular filters increases the amount of light passing through from 0% to 25%.', 'duration': 83.264, 'highlights': ['Introducing a third filter at a 45 degree angle between two perpendicular filters increases the amount of light passing through from 0% to 25%.', '50% of the photons passing through the vertical filter will also pass through the diagonal filter, and once they do, they have a 50-50 chance of passing through the filter oriented at 90 degrees.', 'The experiment becomes all the weirder when specific probabilities for angles between 0 and 45 degrees are considered.']}, {'end': 1318.748, 'start': 1027.832, 'title': 'Quantum polarization and wave-particle duality', 'summary': 'Discusses the quantum nature of light, demonstrating how quantized light behaves probabilistically and emphasizing the wave-particle duality, with key points including the probabilities of photon polarization and the wave nature of quantum mechanical states.', 'duration': 290.916, 'highlights': ["Photons are described as a superposition of oscillating components, with the squares of the amplitudes determining the probabilities of the photon's energy in specific directions.", 'Quantized light behaves probabilistically, with 85% of the time the photon passing through entirely and 15% of the time it gets completely blocked.', 'The need for phase in the superposition of quantum mechanical states arises from their wave nature, with measurements determining the probabilities of finding particles in specific base states.']}], 'duration': 374.18, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MzRCDLre1b4/pics/MzRCDLre1b4944568.jpg', 'highlights': ['Introducing a third filter at a 45 degree angle increases light transmission from 0% to 25%', '50% of photons passing through the vertical filter also pass through the diagonal filter', 'Quantized light behaves probabilistically, with 85% passing through entirely and 15% getting blocked', 'Specific probabilities for angles between 0 and 45 degrees are considered in the experiment', 'The need for phase in the superposition of quantum mechanical states arises from their wave nature']}], 'highlights': ["The energy of electromagnetic waves is quantized in integer multiples of Planck's constant", "The addition of two distinct waves as a valid solution to Maxwell's equations in a vacuum is explained, based on the linearity of the equations", 'The chapter emphasizes the importance of understanding light and wave mechanics in quantum mechanics, particularly the relationship between energy in classical waves and the probabilities governing quantum behavior', 'The video aims to present a topic that sets the right intuitions for anyone interested in learning quantum mechanics, focusing on the late 1800s understanding of light as waves in the electromagnetic field', "Maxwell's equations describe the interaction between electric and magnetic fields, resulting in propagating waves"]}