David Tong > Quotes

“the energies are (2.73) All energies are proportional to”
― 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

“​ Moreover”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“It’s difficult to overstate the importance of the harmonic oscillator in quantum mechanics. It is”
― 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 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

“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

“All of this means that”
― 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

“Solutions to the Schrödinger equation that behave as (2.85) are called bound states because they are necessarily trapped somewhere in the potential.”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“Gaussian wavepacket. Clearly it describes a state that is fairly well localised in space. But”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“The Schrödinger equation is (2.4)”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“Given an operator acting on some class of functions”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“The quantum novelty is that the wavefunction itself is not restricted only to the well: it leaks out into the surrounding region”
― 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

“There are”
― 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

“we take a state with some fixed number of photons”
― 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

“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

“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

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

“Rather than thinking of them as quantum probabilities for a single particle”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“The discreteness in quantities like momentum and energy is one of the characteristic features of quantum mechanics. However”
― 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

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

“In quantum mechanics”
― 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 learn that the Gaussian wavefunction (2.64) that we guessed earlier is actually the lowest energy state of the system”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics

“This means”
― David Tong, Quantum Mechanics: Volume 3: Lectures on Theoretical Physics