[Physics] Schrödinger Equation 1-D particle in a box

quantum mechanicsschroedinger equation

Apologies if this has been asked already

  1. For the 1D particle-in-a-box example, how do we determine the weights of each eigenfunction in the general (time-dependent) solution that fully describes the system? Do we need extra initial conditions for this and if so, what might these conditions be and how might we measure/determine them?

  2. Also, doesn't each eigenstate have constant energy by definition? If so, then shouldn't the momentum of the particle in an eigenstate have a constant magnitude? According to Wikipedia, momentum for the 1D particle-in-a-box has a distribution: https://en.wikipedia.org/wiki/Particle_in_a_box#Position_and_momentum_probability_distributions

  3. Finally, according to QM, $\langle \psi|O|\psi\rangle$ can be used to determine the expectation of observable O. Does QM similarly provide a way to determine the distribution of the observable? If not, it seems the theory is rather incomplete

Thanks in advance

Edit

Thanks for the answers all! A couple edits

  1. Assume psi(0) is unknown, which I'd imagine would generally be the case in an experiment. What would we then expect the general wavefunction to look like? Are all possible coefficient vectors (c1,c2,…,cn) equally likely in practice, or are some combinations more likely than others?

  2. Momentum of particle in a box explains it – QM is incompatible with de Broglie's matter wave theory apparently. Good to know. Also although energy states are quantized, this does not appear to be the case for kinetic energy.

  3. symanzik138 and freecharly answered this one; it makes sense now.

Best Answer

1) A time dependent state can be written as a superposition of the eigenstates of the Hamiltonian. I am assuming that the Hamiltonian is a constant one; does not change with time. The coefficients or weights of the eigenstates are complex. so, for any state $\psi$,

$\psi(t) = c_1 \phi_1 + c_2 \phi_2 + \dots $

where $\phi_i$ are the normalized eigenstates of the Hamiltonian. The coefficients

$c_i(t) = \langle \phi_i|\psi(t) \rangle$

are time dependent.

And the time dependence has the precise form:

$c_i(t)=\exp(-\imath E_i t/\hbar) c_i(0)$

where $E_i$ is the energy of the state $\phi_i$. Thus given the initial coefficients $c_i(0)$, you can determine the coefficients $c_i(t)$ and therefore the state at any other instant $t$. Equivalently you can take $\psi(0)$ also as the initial condition, as $c_i(0)$ can be inferred from $c_i(0)=\langle \phi_i|\psi(0) \rangle$.

2) Constant energy does not mean a constant momentum. This is because the box is not translation invariant. However, the magnitude of the momentum is a constant (in the case of the simple 1D quantum well. This is not true in general). You can in fact write the energy eigenstate as a linear combination of states of momenta $p$ and $-p$. So momentum distribution is nonzero at $\pm p$ only.

3) Distribution of an observable can be interpreted as follows. The probability of finding a value of $o$ for a measurement (say position or momentum) on a system in $\psi$. Such measurements are associated with a corresponding operator $O$. $p(o) = |\langle\psi | \phi_o\rangle|^2$ if $o$ is an eigenvalue of $O$, and $p(o)=0$ otherwise. $\phi_o$ is the normalized eigenstate of $O$ withe eigenvalue $o$. (I am assume no degeneracy).

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