Observables of position and momentum have a continuous spectrum

1. Dec 3, 2005

asdf1

could someone explain this paragraph taken from "concepts of modern physics" by arthur beiser pg175? i'm having trouble understanding it...

"A dynamical variable G may not be quantized. In this case, measurements of G made on a number of identical systems will not yield a unique result but instead a spread of values whose average is the expectation value
<G>=(integrate) G(psi^2)dx"

and why if the electron's position in the hydrogen atom isn't quantized, we have to think of the electron in the vicinity of the nuvleus with a ceratian probability?

2. Dec 3, 2005

Galileo

Some physical observables can take on discrete (quantized) values, like the energy of a particle in an infinite potential well, harmonic oscillator, or the energy of the electron in an hydrogen atom. In this case the eigenvalue-spectrum of the corresponding observable is discrete. This is not always the case though. The observables of position and momentum have a continuous spectrum (ie, not quantized).
The expectation value is calculated the same way as with any observable:
$$<G>=<\psi|G|\psi>$$

So the position of the electron in a hydrogen atom in not quantized (it's after all described by a continuous wavefunction) and thus given by a probability density |psi|^2

3. Dec 3, 2005

asdf1

thank you very much for explaining!!!：）