27 votes
Accepted

How should we think about Spherical Harmonics?

How should we think about Spherical Harmonics? In short: In the same way that you think about plane waves. Spherical harmonics (just like plane waves) are basic, essential tools. As such, they are ...
20 votes
Accepted

What is the difference between a Hilbert space of state vectors and a Hilbert space of square integrable wave functions?

This is a good question, and the answer is rather subtle, and I think a physicist and a mathematician would answer it differently. Mathematically, a Hilbert space is just any complete inner product ...
  • 42.6k
17 votes
Accepted

How is Pauli's exclusion principle valid for electrons of two hydrogen atoms in ground state, having same spin?

A quantum state includes the information about a particle's position. Two particles with the same quantum numbers at different locations are in different states, so are allowed by the exclusion ...
  • 42.6k
17 votes
Accepted

Why can't every quantum state be expressed as a density matrix/operator?

The statement by yuggib is correct. To put it in perspective, I'll start with a completely general formulation, and then I'll show how vector-states and density operators fit into that picture. I won'...
17 votes
Accepted

Quantum mechanics and rigorous math

Kets In quantum mechanics the possible states of the system are elements of a separable, projective Hilbert space (i.e. two states differing by an overall complex constant are equivalent). The kets e....
  • 6,487
14 votes

Bra ket notation rigorous way

Consider the case of a vector space of countable dimension, with some orthonormal set of basis kets $\left\{\vert\mathbf{e}_i\rangle\right\}$. The orthonormality condition is stated as $\langle \...
  • 3,410
14 votes
Accepted

Are two states with the same measurement probabilities necessarily equal up to unitary equivalence?

Let $H$ denote a finite-dimensional complex Hilbert space, $\rho$ and $\rho^\prime$ be two density matrices, i.e. positive semi-definite operators with unit trace and $A$ an arbitrary hermitian ...
14 votes
Accepted

Can the wavefunction be inferred from the expectation values of operators?

It actually suffices to know the expectation values of all projection operators of the form $P_\psi:=|\psi\rangle\langle \psi|$ for $\psi \in H$ (which are of course observables). Indeed, suppose we ...
13 votes

The definition of Gaussian State

The most general description of a quantum system is given by a density matrix $\rho$. It has dimensions of $N \times N$, where $N$ is the number of degrees of freedom of the system: 2 for a 2 level ...
  • 842
13 votes
Accepted

The definition of Gaussian State

A Gaussian state is a ground or thermal state of a (bosonic or fermionic) Hamiltonian which is quadratic in the creation and annihiliation operators. Those states are fully characterized by ...
12 votes

Is a wave function a ket?

The definition is $$ \psi(x)=\langle x| \psi\rangle, ~~~\leadsto \\ |\psi\rangle= \int dx ~~\psi(x) | x\rangle , ~~\leadsto \\ |\Psi\rangle= \int dx ~~ A e^{-ikx}| x\rangle . $$ Wavefunctions are ...
11 votes
Accepted

Quantum Joke (not a real joke, not a riddle)

The signs are important for fixing an out of order machine. Define the states $|\pm\rangle$ as: $$|\pm\rangle = \frac{1}{\sqrt{2}}\left[\left |\text{Working}\right\rangle\pm \left |\text{Down}\right\...
  • 9,698
11 votes

How should we think about Spherical Harmonics?

How familiar are you with Fourier analysis? For instance, if you want a complete basis of functions defined on the unit square, you would pick: $$ |n, m\rangle \equiv e^{2\pi i nx}e^{2\pi i my}$$ with ...
  • 24.7k
11 votes
Accepted

What do the off-diagonal elements of Hamiltonian matrix physically represent?

Remember, the meaning of the Hamiltonian in the first place is that it generates time translations via the Schrodinger equation: $$ i \hbar \frac{\partial}{\partial t} |\psi(t) \rangle = \hat{H} | \...
  • 2,429
10 votes
Accepted

What's the difference between classical and quantum vector superposition?

This is all just a result of sloppy language on the part of people describing quantum mechanics. The state $$ \left\lvert \Psi \right\rangle = \frac{1}{\sqrt{2}} \left( \left\lvert \uparrow \right\...
  • 23.1k
10 votes
Accepted

Uniqueness of bra-ket correspondence

Two complex topological vector spaces $X$ and $Y$ are said to be in duality if there is a sesquilinear map $$b:X\times Y\to \mathbb{C}\; .$$ The idea is that, given such map, a dual action of $X$ on $...
  • 11.6k
10 votes
Accepted

Does every ray in the Hilbert space of a system correspond to one and only one physical state and vice versa?

Short answer: No. Long answer: The concept of state "pre-exists" the concept of Hilbert space, in the following sense. The starting point for the mathematical description of a physical system is the ...
  • 11.6k
10 votes
Accepted

Riesz Representation Theorem and Inner Products

No, $\langle x|$ is not $L^2$ continuous so that the Riesz theorem does not apply. Actually it is not even a functional on that space since the vectors are equivalence classes up to zero Lebesgue ...
10 votes
Accepted

What are the necessary and sufficient conditions for a wavefunction to be physically possible?

If you want to use the theory of probability, a necessary condition for a wavefunction to be physically meanigful is $$\psi \in L^2(\mathbb{R}^3,d^3x)\:.$$ That is because, as a basic postulate of QM, ...
10 votes
Accepted

Measurement on mixed states

As mentioned by the OP both versions are the same. For an observable $A$ of the form $$A = \sum\limits_k a_k \, P_k \quad , $$ with the projections $P_k^2 =P_k = P_k^\dagger$ on the eigenspace ...
10 votes

Are two states with the same measurement probabilities necessarily equal up to unitary equivalence?

Actually $\rho=\rho'$. Indeed, specializing $A= P= |\psi\rangle \langle \psi|$ for $||\psi||=1$, the hypothesis implies $\langle\psi| (\rho-\rho')\psi\rangle =0$. Linearity permits to relax the ...
9 votes

How is a bound state defined in quantum mechanics?

The bound state is defined such that the probability density average will be finite at some particular space region when time passes. While for unbounded states, as time passes, the probability ...
9 votes
Accepted

In quantum mechanics, is $|\psi\rangle$ equal to $\psi(x)$?

To start, the kets are vectors, which means if we want an explicit realization of them, we would need to write them with respect to some basis. The first basis most people see is the position basis, ...
9 votes

Why do wavefunctions for stationary states include $e^{-iEt/\hbar}$?

The time dependent Schroedinger equation looks like this: $$ i\hbar \frac{\partial \Psi}{\partial t} = H \Psi = \left ( -\frac{\hbar^2 }{2 m}\frac{\partial^2}{\partial x^2} + V(x,t) \right ) \Psi(x,t)...
  • 23.1k
9 votes

Very precisely explaining when phase plays a role or doesn't play a role in QM

Basis vs not-basis is not the relevant distinction, because any vector can be part of a basis. However, what I think you're getting at, and you're correct, is that you have to separate the physical ...
  • 26.1k
9 votes
Accepted

What will happen if I multiply a ket vector by a complex number?

The states of a quantum system, the kets, are elements of a complex Hilbert space (modulo a phase). A complex Hilbert space is nothing more than a fancy (in)finite dimensional vector space equipped ...
  • 4,285
9 votes

How should we think about Spherical Harmonics?

The Spherical Harmonics are a complete orthonormal basis for the space of the functions defined on the sphere. This statement means: You can decompose any arbitrary function $f(\theta,\phi)$ defined ...
9 votes
Accepted

An extension of von Neumann entropy to observables

Let $\rho=p\lvert a\rangle\langle a\rvert+(1-p)\lvert b\rangle\langle b\rvert$ , with $\lvert a\rangle$ and $\lvert b\rangle$ two orthogonal maximally entanged states. Then, the reduced density ...
8 votes
Accepted

How is a bound state defined in quantum mechanics?

Bound states are usually understood to be square-integrable energy eigenstates; that is, wavefunctions $\psi(x)$ which satisfy $$ \int_{-\infty}^\infty|\psi(x)|^2\text dx<\infty \quad\text{and}\...
8 votes

Is the partial trace of a mixed state always mixed? If not, are there natural examples where the partial trace of a mixed state is a pure state?

Start with a pure qubit and a mixed qubit, then trace out the mixed qubit. The overall state goes from mixed to pure.
  • 5,934

Only top scored, non community-wiki answers of a minimum length are eligible