- #1
etotheipi
As a simple example, the probability of measuring the position between [itex]x[/itex] and [itex]x + dx[/itex] is [itex]|\psi(x)|^{2} dx[/itex] since [itex]|\psi(x)|^{2}[/itex] is the probability density. So summing [itex]|\psi(x)|^{2} dx[/itex] between any two points within the boundaries yields the required probability.
The integral I'm confused about is the expansion of the quantum state vector, like so$$\left| \psi \right> = \int \psi(x) \left| x \right> dx$$If the wavefunction were defined between 0 and 1, and I were to take the left Riemann sum (not rigorously, but just vaguely for explanation purposes), I would get $$\left| \psi \right> = \psi(0) \left| 0 \right> dx + \psi(dx) \left| dx \right> dx + \psi(2dx) \left| 2dx \right> dx + ...$$What is the purpose of the trailing [itex]dx[/itex] in each term (i.e. which part does it multiply to), or is this the completely wrong way to think about it?
The integral I'm confused about is the expansion of the quantum state vector, like so$$\left| \psi \right> = \int \psi(x) \left| x \right> dx$$If the wavefunction were defined between 0 and 1, and I were to take the left Riemann sum (not rigorously, but just vaguely for explanation purposes), I would get $$\left| \psi \right> = \psi(0) \left| 0 \right> dx + \psi(dx) \left| dx \right> dx + \psi(2dx) \left| 2dx \right> dx + ...$$What is the purpose of the trailing [itex]dx[/itex] in each term (i.e. which part does it multiply to), or is this the completely wrong way to think about it?