How do we interpret the wave function and how do extract information from the it?

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The wave function, at a particular time, contains all the information that anybody at that time can have about the particle.  But the wave function itself has no physical interpretation.  It is not measurable.  However, the square of the absolute value of the wave function has a physical interpretation.  We interpret |ψ(x,t)|2 as a probability density, a probability per unit length of finding the particle at a time t at position x.

The probability of finding the particle at time t in an interval ∆x about the position x is proportional to |ψ(x,t)|2∆x.

This interpretation is possible because the product of a complex number with its complex conjugate is a real, non-negative number.  For the probability interpretation to make sense, the wave function must satisfy certain conditions.  We should be able to find the particle somewhere, we should only find it at one place at a particular instant, and the total probability of finding it anywhere should be one.  We therefore require:

In-class activity: Normalize a wave function using a spreadsheet


The most widely used interpretation of the wave function in quantum mechanics is given below:

Quantum mechanics is our current model of the microscopic world.  Like all models, it is created by people for people.

So quantum mechanics does not really describe the system, but the information that the rest of the world can possibly have about the system.

The wave function has no direct physical meaningIt is just one way of storing information.  It stores all the information available to the observer about the system.  To make predictions about the outcome of all measurements, at any time, one has to "do" something to the wave function to extract information.  One has to perform some mathematical operation on it, such a squaring it, multiplying it by a constant, differentiating it, etc.  One has to operate on the wave function with some operator.  (The operator is a specific instruction or set of instructions.)  Every observable is associated with its own operator.