johanneh's profile - activity

I want to compute the expectation value of the energy $ \langle H \rangle $ for a quantum system with the wave function $ \psi(x) $. This is done with $$ \langle H \rangle = \int_{-\infty}^{\infty} \psi^* \hat{H} \psi d x $$ where $$ \hat{H} = - \frac{\hbar^2}{2 m} \frac{d^2}{d x^2} + V(x) $$ is the Hamiltonian operator for some potential $V(x)$. How do I express $ \frac{d}{d x} $ in sage if I don't know on which function it will be applied?

I know it is possible to use psi.diff(x), but is there a way to declare the operator so that it automatically does the derivative when it is multiplied with some function?

Is it possible to declare custom units that can be converted to SI base units? For example: if I want to measure time in ticks and I know that one tick is exactly one nanosecond, can I create a unit 'tick' that can later be converted to seconds?

I know that a custom unit can be declared with

sage: sage.symbolic.units.str_to_unit('tick')
tick

I also know that predefined units can be converted, for example by

sage: units.time.minute.convert(units.time.second)
60*second

However, I want to be able to declare a unit that has both its own name and that can be converted to other units.

sage: tick = sage.symbolic.units.str_to_unit('tick')
sage: tick
tick
sage: #(define relation to SI units here)
sage: tick.convert(units.time.second)
(ValueError: Incompatible units)

Is it possible to define a unit ‘tick’ that can be converted to seconds in this manner?