Quantum Mechanics Quotes

“To determine the Heisenberg equation of motion, we need to know commutation relations in the Heisenberg picture. Happily, these are straightforward to compute using the definition (7.37) for how operators evolve. We have (7.42) We see that the commutation relations remain unchanged in the Heisenberg picture, provided that and are evaluated at equal times. To stress this feature, (7.42) is sometimes called the equal time commutator. The fact that commutation relations are unchanged over time is analogous to the statement that inner products are unchanged in the Schrödinger picture. The Heisenberg equation of motion for is then (7.43) while the equation of motion for is”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“It is, of course, quite possible to study quantum systems in which the Hamiltonian does depend on time. For example, you might be prodding or forcing the system in some way. In this case, there’s no conserved energy and you don’t get to just naively exponentiate the Hamiltonian in (7.33) to solve the Schrödinger equation. We’ll see what replaces this in Chapter 11, which is devoted to the study of Hamiltonians with explicit time dependence.”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“We see that time evolution fits nicely within our idea of symmetries, with the Hamiltonian the generator of time translations: it tells the state how to move. The assumption in writing down (7.33) was that does not, itself, depend on time, which is the statement that the laws of physics are invariant under time translation.”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“(7.35) This, of course, is the time-dependent Schrödinger equation.”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“conserved quantities can usually be traced, via Noether’s theorem, to some symmetry.”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“This makes it clear what’s really setting the energy scale of the hydrogen atom: it is the rest mass energy of the electron! This is given by (6.105)”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“This is called the Rydberg. As you can see, the (negative) ground state energy of the hydrogen atom is exactly one Rydberg; the higher excited states have energy . Note that the hydrogen atom has an infinite number of negative energy bound states.”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“the advantage of the fine structure constant is that it’s a dimensionless quantity. As you can see, it includes the expression familiar from the Coulomb force and it should be viewed as the convention-independent way to characterise the strength of electromagnetism. Moreover, it takes a value that’s easy to remember (6.103)”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“The three angular momentum operators do not mutually commute. They obey the relations (6.10) From a physics perspective, this means that a quantum particle cannot have a well-defined angular momentum in all three directions simultaneously!”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“Let’s look at some properties of the Wigner function. First, it is real. (This follows by taking the complex conjugate and changing variables to .) Second, if we integrate over momentum, and use the fact that , we have (5.78) But that’s rather nice: marginalising over momentum gives us , which we know is the probability distribution over position. Moreover, if we have a normalised wavefunction then we know that .”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“Given a wavefunction , the Wigner function is a function over classical phase space, defined by (5.77) We want to think of this as something akin to a probability distribution over phase space. At first glance that seems unlikely because, as we’ve seen, there is a difference between quantum states whose properties are undetermined and classical probability that can be ascribed to ignorance. This is reflected in the fact that and so we can’t ascribe simultaneous values to both observables. And, indeed, it will turn out that it’s not possible to interpret as a classical probability distribution. Nonetheless, it gets close. Let’s look at some properties of the Wigner function. First, it is real. (This follows by taking the complex conjugate and changing variables to .) Second, if we integrate over momentum, and use the fact that , we have (5.78) But that’s rather nice: marginalising over momentum gives us , which we know is the probability distribution over position. Moreover, if we have a normalised wavefunction then we know that .”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“Given a wavefunction”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“All of this means that”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“This is entirely analogous to the fact that for any energy eigenstate of the harmonic oscillator. But we know what we need to do to construct a state of the harmonic oscillator with a (reasonably) well-defined phase: this is precisely the coherent state (5.33)”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“we take a state with some fixed number of photons”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“the harmonic oscillator is how we describe quantum fields.”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“The time-dependent Schrödinger equation is (4.49)”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“We see that all memory of our previous measurement has been erased by the measurement of . There’s no longer any guarantee that we will still get this time around. Instead”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“In the world of the qubit”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“Take two systems. We’ll call them system described by the Hilbert space”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“3.5.3 Ehrenfest Theorem We can look at how the expectation value of some operator changes with time. We’ll assume that the operator itself has no time dependence”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“If we want to write down the analogous quantum Hamiltonian”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“We see that the Gaussian wavepacket is rather special: it saturates the bound from the Heisenberg uncertainty relation. The class of Gaussian wavefunctions”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“There is an algebraic way of formalising whether two observables can be simultaneously measured or whether”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“This means that if we have many systems”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“where is the exponential operator (3.105)”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“the Hamiltonian”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“A projection operator projects any state onto some subspace of . A projection operator is Hermitian”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“An eigenstate of a Hermitian operator obeys (3.66) where is the eigenvalue.”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“Not any old linear operator qualifies as a physical observable in quantum mechanics. We should restrict attention to those operators that are Hermitian.”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics