4. A particle confined within a one-dimensional region 0 ? x...

4. A particle confined within a one-dimensional region 0 ? x ? a can be described by the wave function ?(x, t) = A sin (?x/a) e^{-i?t}. (a) Find the potential energy V?(x). (b) Find the normalization constant A. (c) Calculate the probability of finding the particle in the interval a/4 ? x ? 3a/4.

Transcript

00:01 Basically a particle in a well with a wave function that depends on x and t and it takes the form of a times the sign of pi x over a times e to the minus i omega t and we want to know a couple things one what is the potential energy um of this system well you could plug this into the schrodinger equation this wave function right in one dimension it looks like this so d squared si d x squared plus v sii sii equals and this is the time dependent shorter energy equation and what you'll find is that basically the potential energy is going to be zero inside of the well so what actually what we have is effectively potential energy that is infinite for well let me restate this a potential energy that is zero when x is between zero and a and that is infinite everywhere else so that's our potential energy and the next part is we want to determine the normalization constant a so to do this we just remember that the integral from zero to a of this way function which is you know si star times si integrated over x should equal one so if we do that we what's going to give us something that looks like this a squared times the integral from zero to a of sine squared pi x over a d x the e to the i omega t is going to cancel when we take the complex conjugate multiplied by the real part or i mean the the original way function and so what we're going to get is that this integral without the a included is going to equal a over two and remember this has to equal 1.

02:05 So this implies a is the square root of 2 over a.

02:10 And then lastly, you want to know what's the probability of finding the particle between a over 4 and 3a over 4.

02:23 So to do this, all we do is evaluate this integral from a over 4 to 3a over 4 of psi star times.

02:34 Si integrated over x, right? this is basically our probability density, the si star side part.

02:40 So if we do this, we're going to get something that looks a little bit like this.

02:46 A over four to three, a over four...

Question

4. A particle confined within a one-dimensional region 0 ? x ? a can be described by the wave function ?(x, t) = A sin (?x/a) e^{-i?t}. (a) Find the potential energy V?(x). (b) Find the normalization constant A. (c) Calculate the probability of finding the particle in the interval a/4 ? x ? 3a/4.

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