Tagged Questions

In physics, an operator is almost always either a square matrix or a linear mapping from one space of functions (often on $\mathbb{R}^N$ or $\mathbb{C}^N$) to the same or other like space of functions. Operators serve as *observables* and as *time evolution operators* in Quantum Mechanics. This tag ...

50 views

How are anti-unitary operators applied?

I was reading about anti-unitary operators from Wikipedia. They give an example of an anti-unitary operator: were $K$ is complex conjugate operation. $\sigma_y$ is defined with respect to two ...
289 views

Commuting observables and CSCO's

I've been looking at some basic quantum mechanics all day in an attempt to better my understanding of the subject. While going over the proof that commuting operators are compatible, I started getting ...
114 views

Would $[\hat{Q},\hat{H}]$ correspond to an observable? [closed]

Would $[\hat{Q},\hat{H}]$ correspond to an observable? Where $\hat{Q}$ is an observable and $\hat{H}$ is the Hamiltonian. Surely that would just mean that $[\hat{Q},\hat{H}]$ would commute i.e. = 0?: ...
306 views

81 views

160 views

Proof for Negele and Orland equation (2.34)

The equation (2.34) of Negele and Orland has $$\mathcal H_\text{A}(\hat{\mathbf p},\hat{\mathbf x}) = \frac{1}{2m}\left(\hat {\mathbf p} - \frac e c \mathbf A(\hat{\mathbf x})\right)^2.\tag{2.34a}$$ ...
240 views

How can I tell if the spectrum of an operator in QM is degenerate?

I know that the collection of all the eigenvalues of an operator $\hat{Q}$ is called its point spectrum, and sometimes two or more linearly independent eigenfunctions share the same eigenvalue, and in ...
86 views

Quantum Harmonic Oscillators

I'm having trouble with quantum harmonic oscillators and I'm not sure how to approach these questions: . I'd really like to get my head around these concepts but I'm struggling to understand fully. ...
459 views

How do you find the projection operator onto an eigenspace if you don't know the eigenvector?

I was working on exercise 2.60 of Nielsen-Chuang which is as follows: Show that $\vec{v}\cdot\vec{\sigma}$ has eigenvalues $\pm 1$, and that the projectors onto the corresponding eigenspaces are ...
208 views

Can I Weyl-order the following Hamiltonian?

I am trying to perform a path integral but I am having trouble with the Weyl ordering of my Hamiltonian. The Lagrangian of the system in question is $$L~=~\frac{1}{2}f(q)\dot{q}^2,$$ where $f(q)$ ...
48 views

When trying to see what symmetries an operator generates, how do you “decide” what coordinate to apply it to?

Suppose I have $\hat{O}_{1}=-i\hbar\partial_{x}$ then \begin{eqnarray} e^{-i\gamma\hat{O}_{1}/\hbar}x\,e^{i\gamma\hat{O}_{1}/\hbar}=x+\gamma \end{eqnarray} and \begin{eqnarray} e^{-i\gamma\hat{O}_{...
121 views

Time Evolution of Position Operator

I am trying to understand why $$e^{-it\triangle}xe^{it\triangle}=x-2it\nabla$$ where $x$ is just multiplication operator by $x$. In particular, the text says this can be seen by differentiating with ...
268 views

What exactly implies the need of quantum mechanics for self-adjoint and not only symmetric operators? [duplicate]

We know that quantum mechanics requires self-adjoint operators, not only symmetric. Can we say that this follows ONLY from the two following axioms of quantum mechanics, namely that each observable ...
409 views

Deriving cross product from angular momentum algebra

Is it possible to derive: $$\hat{L}=\hat{r}\times \hat{p}$$ from the angular momentum algebra: [\hat{L}_i,\hat{L}_j]=i\ \hbar\ \epsilon_{ijk}\hat{L}_k\ ? \...
735 views

Does the time ordering operator have a rigorous definition?

In quantum field theory, the time ordering operator (TOO) appears in the formal expressions for the scattering amplitudes. It acts upon a product of operators that each depends on time, and returns ...
357 views

What is the meaning of commuting Hamiltonians?

I have two quantum mechanical Hamiltonians such that $$[\hat{H}_1,\hat{H}_2] = 0,$$ where $\hat{H}_1$ and $\hat{H}_2$ act on the same set of states. What is there to ...
290 views

Tensor product of operators in QM

If I wanted to find the coefficients of a linear transformation between 2 vectors in the basis for 2 spin $1/2$ paticles (let's say for starters we are not even looking for a unitary transform): \...
193 views

43 views

88 views

Quantum state of a system after measurements with non-commutative operators

a) Assume two operators $A$ and $B$. 1) Assume $$[A,B]=0$$ and $$ψ= \sum c_n u_n ~~~~\text a~ wavefunction~ describing~ the~ state~ of~ the~ system$$ with $$Aψ=a_n u_n$$ $$Bψ=b_n u_n$$ If we ...
Using creation and annihilation operators to prove the expression for the $n$th excited state in terms of the vacuum state
How does one prove that the $n^{th}$ excited state of a quantum harmonic oscillator (QHO) can be obtained by applying the creation operator $a^{\dagger}$ $n$-times to the vacuum state $\vert 0\rangle$?...