# Is the space of possible positions of a particle in QM discrete?

I recently updated my understanding about quantum mechanics from popular science level to basic undergraduate level.

What surprised me is that for the quantum state of a particle, the wave function for that state using the position of that particle as a basis, is a function defined on the continuum.

I understand now how there can be a discrete space of energy levels of a particle when it is trapped.

• But when it comes to the position of a particle, does the fact that we represent the state of a particle as a continuous wave function on position space mean that QM states that the position of a particle can potentially be observed anywhere on the continuum?

• similarly, Im not sure how this works for non trapped particles. It seems energy is not quantized there since momentum alao has a wave function defined on the continuum, so does that mean that the space of possible kinetic energy levels for a non trapped particle is not discrete?

• if the answers to these questions are yes, how does this square with my popular science understanding that there is a minimum segment in space, time, and energy levels given by plancks constant?

• – Qmechanic Mar 12 '18 at 19:40

3> The segment you refer to is not in constrast to the previous statements. Planck's constant $\hbar$ always comes into play. For example, if you remember the hamiltonian eigenvalues at the harmonic oscillator: $$E_n=\left(n+\frac{1}{2}\right)\hbar\omega$$
• +1 Good answer. But I think you could improve your answer to number 3. It's not just that $\hbar$ comes into play. The question seemed to be suggesting that "popular" accounts of QM indicate that everything — including space and time — are discrete, and the size of those discrete dimensions are given by the Planck length and Planck time. This site has plenty of good answers to that question, like this answer or this one or this one. – Mike Mar 12 '18 at 18:48