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TIme-independent schrodinger equation

  1. Jul 9, 2012 #1
    I've been looking into the time independent schrodinger equation (E[itex]\Psi[/itex] = Ĥ[itex]\Psi[/itex].)

    I know that [itex]\Psi[/itex] is the wave function and Ĥ is the hamiltonian operator. I know that Ĥ is the total of all the energies in a system. What exactly is the wave function? Is it a quantum state? And what does the E represent?

    Thanks,
     
  2. jcsd
  3. Jul 9, 2012 #2
    The following is for the one-particle case (i.e. a system with one particle).

    Mathematically psi is a function on space, expressed as [itex]\psi(x,y,z)[/itex] and it is complex-valued. Mathematically [itex]\psi : \mathbb R^3 \to \mathbb C[/itex]. Its minimal physical meaning is that its modulus squared, mathematically [itex]| \psi |^2[/itex], gives the probability density of finding a particle at a certain point in space.

    This is true, in a sense, but I think it's prone to misconception. It is true that knowing [itex]\hat H[/itex] is equivalent to knowing all the energy levels of the system (this is known in mathematics as the spectral theorem for operators) but that is not what [itex]\hat H[/itex] is itself. It is an operator meaning that is a function which takes a function like psi as its argument/its input, and gives another such function as its image/output. A very simple Hamiltonian (that is its name) is [itex]\hat H = -\frac{\partial^2}{\partial x^2}[/itex] (ignoring constants and expressing it in the one-dimensional case). As input it takes a function [itex]\psi[/itex] and as output it gives minus its second derivative with respect to x, this is of course a new function from [itex]\mathbb R^3[/itex] to [itex]\mathbb C[/itex].

    E is the specific energy level your system has (in other words you have to choose one; the idea is that you choose the energy level(*) of your system and then calculate what [itex]\psi[/itex] satisfies that equation, and then its modulus squared gives the probability density as discussed above).

    (*) Note that you can't choose just any value for E; the possible list is determined by [itex]\hat H[/itex], as you said.

    EDIT: yes, another name for [itex]\psi[/itex] is the so-called "quantum state". Also called "the quantummechanical wavefunction".
     
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