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I am trying to learn more about quantum mechanics. I am reading a book by Griffiths that I like. I'm trying to summarize what I've learned. So below I provided three assumptions. I'd like to know if they are correct.

Consider a particle in space and time.

  1. We cannot know where the particle is with certainty. If we perform the same measurement experiment on an ensemble of identically prepared quantum systems, on average we may find the particle at one location more often than others.
  2. Under certain conditions, we can know and predict future probability distributions. Specifically, this is when the probability of the particle distribution is constant in time (eg stationary states).
  3. Given a probability amplitude $\Psi(x,0)$, we can predict the future value $\Psi(x,t)$ through the Schrödinger equation.

I'm pretty sure I'm misunderstanding something. In particular, No.3 suggest that I could predict future probability distributions of the wave function and I know from talking to people on SE that's wrong.

My Question:

Can someone explain which (if any) of my assumptions above is wrong and explain why?

Stan Shunpike
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2 Answers2

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We cannot know where the particle is with certainty.

The particle, in general, does not have a definite location to know.

Under certain conditions, we can know and predict future probability distributions.

The evolution of the state is determined by the Hamiltonian (in the Schrodinger picture). The problem is that we don't know the Hamiltonian for the measurement apparatus. Thus, the certain conditions are that the Hamiltonian is known. The "collapse of the wavefunction" is essentially a reflection or our ignorance of the Hamiltonian including the measurement apparatus.

Given a probability amplitude Ψ(x,0), we can predict the future value Ψ(x,t) through the Schrödinger equation.

Correct in principle. The devil is in the details of the time evolution operator.

Alfred Centauri
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  1. Correct, expect just don't say "on average," just we will find the particle at different locations more often than others in accordance with the probability distribution described by a wave function.

  2. Incorrect, you may always calculate the evolution of an initial state if you know the effective Hamiltonian or effective Action the state is subject to. However, it may not be easy or even possible to solve the quantum system analytically i.e. with out the use of an approximation or computer. The reason you see stationary states is because they tend to be easier to solve in closed form, hence the simplest of quantum systems appear in introductory texts.

  3. Correct.

BVPhD
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