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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?

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to answer your questions:

1> Yes, that is the case.

2> Yes, that is the case also. However, I don't really understand what you mean by "the momentum also has a wavefunction". Do you refer to the Fourier transform of the spatial wavefunction.

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: $$E_n=\left(n+\frac{1}{2}\right)\hbar\omega$$

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    $\begingroup$ +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. $\endgroup$ – Mike Mar 12 '18 at 18:48
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    $\begingroup$ 2> yes I mean the fourier transform of the spatial wavefunction. 3> But the harmonic oscillator is only one system. My question is: if it is possible for a partical to have a momentum anywhere on the continuum, and energy is a function of momentum, then it is not the case that there is only a discrete set of possible energy levels, so the idea that there is a minimum possible change in position or energy level (which is an idea I got from popular science articles) would be incorrect. $\endgroup$ – user56834 Mar 12 '18 at 18:52
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In response to your third question, the Plank length might be what you consider the "quantized" space and the Plank time is the "quantized" time. These are fundamental ideas for loop-quantum gravity, but not a result of the Schrodinger equation or quantum mechanics.

In quantum mechanics, the frequency (and thus energy) of a particle's wavefunction are quantized, but it may be found anywhere in space. The probability of finding it in a region is given by the integral of the amplitude squared of the particle's wavefunction over that region.

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