All Questions
24 questions
0
votes
1
answer
84
views
Odd notation $\stackrel{\leftarrow}{\nabla}$ for a gradient
I've tried working out the Heisenberg EOM for the 4-current operator. Two very beautiful articles (DOI: 10.1103/PhysRevA.84.042107, DOI: 10.1103/PhysRevA.90.012508) present this result, but I have not ...
- 51
0
votes
0
answers
28
views
Evaluating the commutator of derivative and position [duplicate]
In Zettili's book on quantum, the fully worked problem 2.6 asks to show
$$
\hat{A} = i(\hat{X}^2+1)\frac{d}{dx} + i\hat{X}.
$$
Is Hermitian. Where $\hat{X}$ is the position operator. I took the ...
1
vote
1
answer
288
views
Question on how to make product rule for differentiation consistent with operators? [duplicate]
By the product rule for differentiation:$$\frac{\partial(\hat A\psi)}{\partial x}=\left(\frac{\partial\hat A}{\partial x}\right)\psi+\hat A\left(\frac{\partial\psi}{\partial x}\right)\tag{1}$$
Where $\...
3
votes
2
answers
267
views
What does $\dot x$ mean as an operator in quantum mechanics?
I've been looking at a paper titled "Feynman's proof of the Maxwell Equations" by Freeman Dyson (American Journal of Physics 58, 209 (1990); https://doi.org/10.1119/1.16188) and I'm confused ...
- 41
0
votes
2
answers
123
views
Ambiguity in Notation for Operators in Quantum Mechanics
Let's say I am trying to find the commutator of operators $\mathbf{A}$ and $\mathbf{B}$, and I get
$$[\mathbf{A},\mathbf{B}]=\nabla^2 f(x,y,z).\tag{0}$$
There seems to be some ambiguity here.
In ...
2
votes
1
answer
115
views
Schrödinger equation: $\frac{\partial}{\partial t}$ and $\frac{d}{dt}$ [duplicate]
I have seen two different forms of Schrödinger equation:
$$i\hbar\frac{\partial|\psi(t)\rangle}{\partial t}=\hat{H}|\psi(t)\rangle$$
and
$$i\hbar\frac{d|\psi(t)\rangle}{d t}=\hat{H}|\psi(t)\rangle.$$
...
- 4,276
-1
votes
1
answer
650
views
What do some of the symbols in the Schrodinger Equation mean? [closed]
The Time Dependent Schrodinger Equation has the form
$$i\hbar\frac{\partial}{\partial{t}}\Psi=-\frac{\hbar^2}{2m}\left(\nabla^2+V\right)\Psi$$
and the Time Independent Schrodinger Equation has the ...
- 2,021
1
vote
4
answers
420
views
What do $\nabla$ and $\frac{d }{d t}$ mean when they are by themselves?
In QM and QFT, I have seen some equations where they have just the derivative and/or the gradient without specifying what it is acting on.
Taken from wiki.
This does not make sense to me since I ...
- 2,042
7
votes
2
answers
652
views
Issue in deriving Ehrenfest's theorem
Working in Schrodinger picture, while deriving Ehrenfest's theorem, we go -
$$
\frac{d}{d t}\langle A\rangle=\frac{d}{d t}\langle\psi|\hat{A}| \psi\rangle
$$
$A$ is an operator. Expanding RHS-
$$
\...
- 440
2
votes
3
answers
109
views
Small doubt on derivatives acting on kets/bras
I have a quick, silly question. If $\psi(x):=\langle x|\psi\rangle$, does the bra $\langle x|$ 'go through' the $\partial_x$ operator, as in $$\langle x|\partial_x|\psi\rangle=\partial_x\psi(x) \quad ?...
- 89
8
votes
1
answer
999
views
Two different versions of Schrödinger's equation - are they equivalent?
For simplicity, let's look at the case of one particle in one dimension. We usually think of the wave function as a function
\begin{align}
\Psi\colon\mathbb R\times[0,\infty[&\to\mathbb C\\
(x,t)&...
- 1,911
14
votes
3
answers
1k
views
What is meant by a partial derivative of a ket?
In my QM book I often see partial derivatives mixed with kets, like
$$
\frac{\partial}{\partial a} |\psi \rangle
$$
where $a \in \{x, y, z\}$. Here I'm assuming that $| \psi \rangle \in \mathbb{C}^n$ ...
- 337
2
votes
1
answer
913
views
Expectation value of time derivative of operator vs. time derivative after operator
Problem 3.18 in Griffiths's Introduction to Quantum Mechanics (3rd ed.) asks to apply the generalised Ehrenfest theorem to operators like the Hamiltonian and momentum operator. The purpose of the ...
- 357
4
votes
3
answers
263
views
What does it mean "differentiation with respect to the coordinates of particle 1 or 2"?
I was reading Introduction to Quantum Mechanics by Griffiths. In Chapter 5, Identical Particles, I came across the notation $\nabla_1$ and $\nabla_2$. Griffiths writes that it means "differentiation ...
-1
votes
1
answer
58
views
Expectation of partial time derivatives of $x$ in QM
In Ehrenfest theorem we know that
$$m\frac{d\left< x\right>}{dt}=\left< p\right>+m\left<\frac{\partial x}{\partial t}\right>.$$
So how can I exactly calculate a specific $\left<\...
- 11
2
votes
1
answer
425
views
Question about commutators acting on wavefunctions
Consider a commutator acting on a 1D wavefunction:
$$[\frac{\hbar}{i} \frac{d}{dx},x]\psi(x)=(\frac{\hbar}{i} \frac{d}{dx}x-x\frac{\hbar}{i} \frac{d}{dx})\psi(x).$$
Now does this mean
$\frac{\hbar}{...
- 4,276
0
votes
1
answer
105
views
Operator $A$ only act on the neighboured state or operator but not the entire expression?
In state vector formalism $A|\psi(x)><u(x)|=(A|\psi(x)>)<u(x)|$, where $A$ only act on $|\psi(x)>$
However, in terms of wave formalism, suppose $A$ is the well known $\frac{d}{dx}$.
...
- 3,306
4
votes
2
answers
758
views
Ordinary vs. partial derivatives of kets and observables in Dirac formalism
I'm a bit confused as to when ordinary and partial derivatives are used in the Dirac formalism.
In the Schrödinger equation, for instance, Griffiths [3.85] uses ordinary derivatives:
$$ i \hbar \...
- 380
0
votes
2
answers
2k
views
Gradient of a wave function, notational confusion
I'm reading from "Quantum Physics for Dummies", by Steven Holzner. In chapter two, entitled "Entering the Matrix: Welcome to State Vectors", the author introduces the notation for a gradient of a wave ...
- 768
0
votes
1
answer
1k
views
Partial Derivative and Dirac Notation [duplicate]
Does the partial derivative of $\langle x'|\alpha\rangle$ with respect to $x'$ equal $|\alpha\rangle$? Why?
Note: $|\alpha\rangle$ is an arbitrary ket, $x'$ is an eigenvalue, and $\langle x'|$ is an ...
- 243
0
votes
1
answer
187
views
Density operator as a function of time
Given the density operator $\rho = \sum_iw_i | \alpha^{i} \rangle \langle \alpha^{i}|$, how does the density operator change with time. Apparently I should get $$i \hbar \frac{\partial \rho}{\partial ...
- 1,053
1
vote
3
answers
301
views
Why is $\frac{d^2}{dx^2}=\left(\frac{d}{dx}\right)^2$ justified in the equation for the square of the momentum operator?
The square of the momentum operator $\hat p$ from the time independent Schrödinger equation is $$\hat p^2=-\hbar^2\frac{d^2}{dx^2}\tag{1}$$ in the one dimensional case.
So if we solve this equation ...
- 2,520
18
votes
2
answers
16k
views
Do derivatives of operators act on the operator itself or are they "added to the tail" of operators?
How do derivatives of operators work? Do they act on the terms in the derivative or do they just get "added to the tail"? Is there a conceptual way to understand this?
For example: say you had the ...
- 1,156
1
vote
2
answers
319
views
Notation for differential operators and wave function math
I know that $[\frac {d^2}{dx^2}]\psi$ is $\frac {d^2\psi}{dx^2}$ but what about this one $[\frac {d^2\psi}{dx^2}]\psi^*$? Is it this like $\frac {d^2\psi\psi^*}{dx^2}$ or this like $\frac {\psi^*d^2\...
- 11