All Questions
9 questions
8
votes
2
answers
1k
views
What does $\exp\left( ax\frac{d}{dx} \right)$ do on $\psi(x)$?
I'm trying to find out
$$\exp\left(ax\frac{d}{dx}\right)\psi(x)= \ \ ? $$
I tried spending the exponential and then operating the derivatives one by one but I found no pattern. Besides, it gets ...
- 12.1k
4
votes
2
answers
419
views
What does it mean when we say 'The difference between two quantities is of first order'?
This question is about the explanation below Eq.(6.19) of Modern Quantum Mechanics by Sakurai Nepolitano (2nd edition)
Let ${\bf j}(dx)$ be an operator that translates a point $x$ to $x+dx$.
jf(x) = ...
- 397
3
votes
1
answer
1k
views
Taylor expanding a function of an operator?
I am trying to understand the following description in my quantum mechanics textbook:
Let $F(\hat{A})$ be a function of an operator $\hat{A}$. If $\hat{A}$ is a linear operator, we can Taylor expand $...
- 273
2
votes
1
answer
109
views
$x$-derivative of the wave function and its conjugate [closed]
I saw that in order to show that the normalisability of a wave function does not depend on time, there is a necessary step in the calculation that says that:
$$\left(\Psi^*\frac{\partial^2\Psi}{\...
1
vote
2
answers
138
views
The treatment of infinitesimal quantities [duplicate]
Please be advised that my question is different from some of the existing threads like this one.
I have long been convinced that if we are to question the value of something which we ultimately are ...
- 862
1
vote
1
answer
288
views
Question on how to make product rule for differentiation consistent with operators? [duplicate]
By the product rule for differentiation:$$\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}$$
Where $\...
1
vote
0
answers
75
views
Wavefunction from the Hamilton-Jacobi formalism [closed]
I was reading this paper, Wavefunctions and Hamilton-Jacobi Equation by Sabrina Pasterski. The author started with the Hamiltonian formalism and then came up with a connection to the Schrodinger ...
- 11
0
votes
0
answers
34
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Wavefunctions and Hamilton-Jacobi equation [duplicate]
I was reading this paper, Wavefunctions and Hamilton-Jacobi Equation. The author started with the Hamiltonian formalism and then came up with a connection to the Schrodinger equation. There was a ...
- 11
-1
votes
1
answer
143
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Quantum derivatives: quantum calculus and their classical limit
Jackson derivatives and their $q,p$-version are defined to be
$$D_qf=\dfrac{f(qx)-f(x)}{(q-1)x}$$
$$D_{q,p}f=\dfrac{f(qx)-f(px)}{(q-p)x}$$
When trying to go $q\rightarrow 1$ and $q\rightarrow p$ I ...
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