9 votes
Accepted

Schrödinger Equation Energy Requirement $E \geq V_{\min}$

It is just matter of math. Even if one may also try to find physical interpretations of this result a posteriori. Let us focus on the identity $$\frac{d^2 \psi}{dx^2}=\frac{2m}{\hbar^2}[V(x)-E]\psi\:.$...
5 votes

What to understand by $\langle \phi | \hat{A}|\psi \rangle$?

Both. Recall that a linear map $A:V\to W$ automatically induces a map $A^*:W^*\to V^*$ where $V^*$ is the dual space of $V$. Given $f\in W^*$ we can evaluate $f(A(x))$ for any $x$ in $V$, and so $f(...
  • 43.3k
3 votes

Schrödinger Equation Energy Requirement $E \geq V_{\min}$

Think about a similar classical problem to understand this. Suppose you have a mass m that can be sitting on the ground anywhere along the $x$ axis. Suppose each point along the axis is at a different ...
  • 28.2k
2 votes

First order state correction for time independent perturbation theory

If $\langle n_0|n_1\rangle$ is imaginary, it can be written as $re^{i\theta}$. If $|n_0\rangle$ solves $\hat H_0|n_0\rangle=E_0|n_0\rangle$, then so does $e^{i\theta}|n_0\rangle$. Choosing instead ...
2 votes

First order state correction for time independent perturbation theory

Let's see what happens if $\langle n^{(0)}|n^{(1)}\rangle$ is not real. Take $\hat{H}_0 \psi^{(0)}(x)=E^{(0)}\psi^{(0)}(x)$ and (to the linear order in $\lambda$) $$(\hat{H}_0+\lambda\hat{V})(\psi^{(0)...
  • 214
2 votes

Does each vector in $su(3)$ represent a different kind/type of gluon (infinite kinds/types of gluons); or, are they all considered the same kind/type?

There are two "levels" of charge in gauge theories. There is the representation, which is a vector space, and then there is a vector in that space. The vectors within a representation are ...
  • 19.2k
2 votes
Accepted

On the separability definition of mixed states

As suggested, I put here as an answer. The idea is that if the goal is to check for separability, there is no loss of generality since by spectral decomposition any density matrix $\rho^{(i)}_A$ (...
  • 1,518
2 votes

Mathematical definition of annihilation and creation operators

$\eta$ is subscripted by $i_\ell$ because the indices $i_1,...,i_{n+1}$ are given on the left hand side and we are summing over their sub indices Talagrand gives on p. 79 the example $$\tag{A} A^\...
  • 1,174
1 vote

Physical difference between $\vert S=0, m = 0 \rangle$ and $\vert S=1, m = 0 \rangle$?

In context of a two spin $\frac{1}{2}$ particle systems, we know that, $\vert S=0, m = 0 \rangle = \frac{1}{\sqrt2}(\vert\uparrow\downarrow\rangle - \vert\downarrow\uparrow\rangle)$ $\vert S=1, m = ...
  • 9,375
1 vote
Accepted

What is the relation between purity and ${\rm Tr}(\rho^2)$, for a density matrix?

I would like to understand the equivalence for a state $\psi$ to be pure and its density matrix $\rho=|\psi\rangle\langle\psi|$ having the property $$\operatorname{trace} \rho^2=1.$$ The density ...
  • 9,375

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