A particle of mass sits in a one-dimensional square well with infinitely high walls and width . The particle is in a 50–50 mixture of states, half in the ground state and half in the first excited state. I want to show how to derive a formula for the complete, time-dependent wave-function of the particle.
I begin by the using the normalizing condition, that . This is because the probability of finding the particle somewhere in space must equal at all times. Because the particle is in a mixture of states, my wave function will take the form . Combining this with the normalizing condition, I get the equation . The individual wave-functions will take the forms and . I can then plug these two equations in the the above normalizing condition for a particle in mixed states, convert all sine functions to cosine functions and cancel out like terms until I get down to the simple expression that which implies that . Now I can use this constant to write down my mixed wave-function which will look as follows . Now, I want to show that the probability of finding the particle between positions and , as measured from the left-hand side of the well, as a function of time, is . This is done by finding the probability amplitude which is the square modulus of the wave function. This will look as follows . After filling in each wave function and multiplying out each term I obtain . After converting the sine terms to cosines and converting the exponentials to trigonometric functions, I can plug in for , which will give me and which is what I wanted to show.