A particle is in an infinite square well of width L and is in the n=3 state. What is the probabi

A particle is in an infinite square well of width L and is...

Key Concepts

Infinite Square Well

The infinite square well is a prototypical quantum mechanical model in which a particle is confined to a finite region of space by infinitely high potential barriers. This model allows for derivation of quantized energy levels and corresponding stationary states, providing a clear example of how spatial confinement leads to discrete energy spectra.

Stationary States and Quantum Numbers

In systems like the infinite square well, solutions to the Schrödinger equation come in the form of stationary states characterized by quantum numbers, which label the different energy eigenstates. These quantized states indicate that even in bounded regions, a particle can only occupy specific energy levels.

Probability Density and the Born Interpretation

The probability of finding a particle within a given region is determined by the square of the magnitude of its wave function, in accordance with the Born rule. This fundamental principle of quantum mechanics links the mathematical description of the wave function to measurable physical outcomes.

Integration over a Spatial Interval

Calculating the probability of finding a particle in a specific spatial region involves integrating the probability density (the square of the wave function) over that region. This process, which must account for proper normalization, is essential for comparing theoretical predictions with experimental measurements.

Transcript

0:00 Hi there.

00:01 So for this problem, we are told that we have a particle in that is in an infinite square well width and is in the state ln equals to 3.

00:14 So what we need to calculate in this is the probability that when it is observed, the particles found to be in the right most 10 % of the well.

00:30 So we need to calculate the probability that we just call with the with the capital letter grid pi.

00:41 So the wave function for an infinite square well is derivide in the test.

00:49 And for a weld of width with a width, we are going to have, of course, with the state n equals to three.

01:03 The wave function inside the well is going to be equal to the square root of 2 over the length lt times the sign of 3 times pi times x divided by lt.

01:20 So the probability that the particle is found in the right most 10 % of the weld is given by the probability, in this case, is going to be the integral from 0 .9 times the length l to the total length l.

01:38 Because this in here corresponds to l -t minus 10 % of lt, which is equal to l minus 0 .1l.

01:48 That's where that comes from.

01:52 And we will have in here that this is the wave function to the squared, because we are taking the probability of this, and we now substitute what is the wave function.

02:03 So we can take out everything that is constant, so we can take out two over l, the integral from 0 .9 l times to the alt, sine to the square of 3 times pi times x divided by l, and this times the differential dx.

02:24 Now, what we can do in here is to use the following identity, that is that two times the sine to the square of theta is equal to one minus cosine of two times theta.

02:42 So we can use this to simplify the integral.

02:46 So we will have that...

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