# Quantum Mechanics  ## Metadata - Author: [[Leonard Susskind]] - Full Title: Quantum Mechanics - Category: #books ## Highlights - The phenomenon of entanglement is the essential fact of quantum mechanics, the fact that makes it so different from classical physics. It brings into question our entire understanding about what is real in the physical world. Our ordinary intuition about physical systems is that if we know everything about a system, that is, everything that can in principle be known, then we know everything about its parts. If we have complete knowledge of the condition of an automobile, then we know everything about its wheels, its engine, its transmission, right down to the screws that hold the upholstery in place. It would not make sense for a mechanic to say, “I know everything about your car but unfortunately I can’t tell you anything about any of its parts.” But that’s exactly what Einstein explained to Bohr—in quantum mechanics, one can know everything about a system and nothing about its individual parts—but Bohr failed to appreciate this fact. I might add that generations of quantum textbooks blithely ignored it. ([Location 64](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=64)) - Tags: [[blue]] - Ordinarily, we learn classical mechanics first, before even attempting quantum mechanics. But quantum physics is much more fundamental than classical physics. As far as we know, quantum mechanics provides an exact description of every physical system, but some things are massive enough that quantum mechanics can be reliably approximated by classical mechanics. That’s all that classical mechanics is: an approximation. From a logical point of view, we should learn quantum mechanics first, but very few physics teachers would recommend that. ([Location 130](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=130)) - Tags: [[blue]] - Our sensory organs are simply not built to perceive the motion of an electron. The best we can do is to try to understand electrons and their motion as mathematical abstractions. ([Location 155](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=155)) - Tags: [[blue]] - indeed the classical and quantum worlds have some important things in common. Quantum mechanics, however, is different in two ways:       1.  Different Abstractions. Quantum abstractions are fundamentally different from classical ones. For example, we’ll see that the idea of a state in quantum mechanics is conceptually very different from its classical counterpart. States are represented by different mathematical objects and have a different logical structure.       2.  States and Measurements. In the classical world, the relationship between the state of a system and the result of a measurement on that system is very straightforward. In fact, it’s trivial. The labels that describe a state (the position and momentum of a particle, for example) are the same labels that characterize measurements of that state. To put it another way, one can perform an experiment to determine the state of a system. In the quantum world, this is not true. States and measurements are two different things, and the relationship between them is subtle and nonintuitive. ([Location 159](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=159)) - Tags: [[blue]] - even a specific type of particle, such as an electron, is not completely specified by its location. Attached to the electron is an extra degree of freedom called its spin. Naively, the spin can be pictured as a little arrow that points in some direction, but that naive picture is too classical to accurately represent the real situation. The spin of an electron is about as quantum mechanical as a system can be, and any attempt to visualize it classically will badly miss the point. ([Location 172](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=172)) - Tags: [[blue]] - The isolated quantum spin is an example of the general class of simple systems we call qubits—quantum bits—that play the same role in the quantum world as logical bits play in defining the state of your computer. ([Location 178](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=178)) - Tags: [[blue]] - The quantum mechanical notation for the statistical average of a quantity Q is Dirac’s bracket notation ⟨Q⟩. ([Location 285](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=285)) - Tags: [[blue]] - What we are learning is that quantum mechanical systems are not deterministic—the results of experiments can be statistically random—but if we repeat an experiment many times, average quantities can follow the expectations of classical physics, at least up to a point. ([Location 294](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=294)) - Tags: [[blue]] - Every experiment involves an outside system—an apparatus—that must interact with the system in order to record a result. In that sense, every experiment is invasive. ([Location 298](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=298)) - Tags: [[blue]] - In quantum mechanics, the situation is fundamentally different. Any interaction that is strong enough to measure some aspect of a system is necessarily strong enough to disrupt some other aspect of the same system. Thus, you can learn nothing about a quantum system without changing something else. ([Location 304](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=304)) - Tags: [[blue]] - One might say that measuring one component of the spin destroys the information about another component. In fact, one simply cannot simultaneously know the components of the spin along two different axes, not in a reproducible way in any case. There is something fundamentally different about the state of a quantum system and the state of a classical system. ([Location 316](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=316)) - Tags: [[blue]] - in this example, the inclusive or is not symmetric. The truth of (A or B) may depend on the order in which we confirm the two propositions. This is not a small thing; it means not only that the laws of quantum physics are different from their classical counterparts, but that the very foundations of logic are different in quantum physics as well. ([Location 412](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=412)) - Tags: [[blue]] - What about (A and B)? Suppose our first measurement yields σz = +1 and the second, σx = +1. This is of course a possible outcome. We would be inclined to say that (A and B) is true. But in science, especially in physics, the truth of a proposition implies that the proposition can be verified by subsequent observation. In classical physics, the gentleness of observations implies that subsequent experiments are unaffected and will confirm an earlier experiment. A coin that turns up Heads will not be flipped to Tails by the act of observing it—at least not classically. Quantum mechanically, the second measurement (σx = +1) ruins the possibility of verifying the first. Once σx has been prepared along the x axis, another mesurement of σz will give a random answer. Thus (A and B) is not confirmable: the second piece of the experiment interferes with the possibility of confirming the first piece. ([Location 415](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=415)) - Tags: [[blue]] - There is a special class of complex numbers that I’ll call “phase-factors.” A phase-factor is simply a complex number whose r-component is 1. ([Location 460](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=460)) - Tags: [[blue]] - The space of states of a quantum system is not a mathematical set;6 it is a vector space. Relations between the elements of a vector space are different from those between the elements of a set, and the logic of propositions is different as well. ([Location 467](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=467)) - Tags: [[blue]] - The vector spaces we use to define quantum mechanical states are called Hilbert spaces. We won’t give the mathematical definition here, but you may as well add this term to your vocabulary. When you come across the term Hilbert space in quantum mechanics, it refers to the space of states. A Hilbert space may have either a finite or an infinite number of dimensions. ([Location 474](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=474)) - Tags: [[blue]] - In quantum mechanics, a vector space is composed of elements |A⟩ called ket-vectors or just kets. ([Location 477](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=477)) - Tags: [[blue]] - a complex vector space has a dual version that is essentially the complex conjugate vector space. For every ket-vector |A⟩, there is a “bra” vector in the dual space, denoted by ⟨A|. ([Location 517](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=517)) - Tags: [[blue]] - The dimension of a space can be defined as the maximum number of mutually orthogonal vectors in that space. ([Location 574](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=574)) - Tags: [[blue]] - Obviously, there is nothing special about the particular axes x, y, and z. As long as the basis vectors are of unit length and are mutually orthogonal, they comprise an orthonormal basis. ([Location 575](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=575)) - Tags: [[blue]] - In classical physics, knowing the state of a system implies knowing everything that is necessary to predict the future of that system. As we’ve seen in the last lecture, quantum systems are not completely predictable. Evidently, quantum states have a different meaning than classical states. Very roughly, knowing a quantum state means knowing as much as can be known about how the system was prepared. ([Location 613](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=613)) - Tags: [[blue]] - variables has a very simple mathematical representation: the space of states for a single spin has only two dimensions. This point deserves emphasis:        All possible spin states can be represented in a two-dimensional vector space. ([Location 649](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=649)) - Tags: [[blue]] - Two orthogonal states are physically distinct and mutually exclusive. If the spin is in one of these states, it cannot be (has zero probability to be) in the other one. This idea applies to all quantum systems, not just spin. But don’t mistake the orthogonality of state-vectors for orthogonal directions in space. In fact, the directions up and down are not orthogonal directions in space, even though their associated state-vectors are orthogonal in state space. ([Location 682](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=682)) - Tags: [[blue]] - States in quantum mechanics are mathematically described as vectors in a vector space. Physical observables—the things that you can measure—are described by linear operators. ([Location 847](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=847)) - Tags: [[blue]] - Observables are the things you measure. For example, we can make direct measurements of the coordinates of a particle; the energy, momentum, or angular momentum of a system; or the electric field at a point in space. Observables are also associated with a vector space, but they are not state-vectors. They are the things you measure—σx would be an example—and they are represented by linear operators. ([Location 850](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=850)) - Tags: [[blue]] - The complex conjugate of a transposed matrix is called its Hermitian conjugate, denoted by a dagger. You could think of the dagger as a hybrid of the star-notation used in complex conjugation and the T used in transposition. ([Location 962](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=962)) - Tags: [[blue]] - We come now to the basic mathematical theorem—I will call it the fundamental theorem—that serves as a foundation of quantum mechanics. The basic idea is that observable quantities in quantum mechanics are represented by Hermitian operators. ([Location 996](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=996)) - Tags: [[blue]] - The Fundamental Theorem        •  The eigenvectors of a Hermitian operator are a complete set. This means that any vector the operator can generate can be expanded as a sum of its eigenvectors.        •  If λ1 and λ2 are two unequal eigenvalues of a Hermitian operator, then the corresponding eigenvectors are orthogonal.        •  Even if the two eigenvalues are equal, the corresponding eigenvectors can be chosen to be orthogonal. This situation, where two different eigenvectors have the same eigenvalue, has a name: it’s called degeneracy. Degeneracy comes into play when two operators have simultaneous eigenvectors, as discussed later on in Section 5.1. One can summarize the fundamental theorem as follows: The eigenvectors of a Hermitian operator form an orthonormal basis. ([Location 999](https://readwise.io/to_kindle?action=open&asin=B00FD36G1Q&location=999)) - Tags: [[blue]]