# Tag Info

### Constant of Motion in Quantum Mechanics for explicit time-dependent Operators

Take any time-independent operator $X$ which commutes with the Hamiltonian and multiply it by an arbitrary real-valued function of time : $$Y(t) = f(t)X$$ Then $Y(t)$ commutes with the Hamiltonian, ...
• 5,771
Accepted

• 20.1k
1 vote

### Where does the complex conjugate term generally come from in a Hamiltonian?

This type of Hamiltonian is a second quantisation one. The most important part is defining your operators $a$ and $a^\dagger$. For example, if you have two interacting dipoles the interaction energy ...
Accepted

### Where does the complex conjugate term generally come from in a Hamiltonian?

The simplest justification is probably that the Hamiltonian has to be Hermitian. After all, its eigenvalues are interpreted as the possible energies of the system, and hence they need to be real. This ...
• 18.1k
Accepted

### Does the Hamiltonian always commute with the Time Evolution Operator?

What can be said about the general case, in which $H$ depends on time explicitly? Specifically: Do $U(t, t_0)$ and $H$ still commute? No, not in general. In general, you can write $U$ as a time-...
• 20.1k
1 vote

The expression for the determinant of a square matrix $M$ of size $N$ is $$\det M = \sum_{\sigma \in S_N} \mathrm{sgn}(\sigma)M_{1,\sigma(1)}\ldots M_{N,\sigma(N)}.$$ In this case, $M_{mn} = A_n\... • 9,524 2 votes Accepted ### Exercise on self-adjointness of Hamiltonian Thanks to @ZeroTheHero who brought me onto the right track, I was able to find the solution to the problem myself. The Hamiltonian with respect to the basis$\{ | \psi \rangle, |\phi \rangle, |\Gamma\...
• 593
In the classical framework, the exchange interaction term S$_{i}$ $\cdot$ S$_{j}$ satisfies the rotation invariance of the x,y,z-direction, but the uniaxial anisotropy term ${(S_{i}^{z})}^2$ breaks ...