A particle of mass m is confined to a box of width L. If the particle is in the first excited st

A particle of mass m is confined to a box of width L. If the...

Key Concepts

Integration over Spatial Regions

To determine the probability of locating a particle within a particular region, the probability density must be integrated over that spatial interval. This integration process connects the theoretical wavefunction to measurable probabilities in experiments.

Probability Density

The probability density is given by the square of the absolute value of the wavefunction. It reflects the likelihood of finding a particle at a specific point in space, and integrating it over a certain region provides the probability of the particle being found there.

Particle in a Box

This is a fundamental quantum mechanics model where a particle is confined within an infinitely deep potential well, meaning it cannot exist outside the box. The model highlights how spatial confinement leads to discrete energy levels and quantized wavefunctions.

Wavefunction

The wavefunction is a complex function that provides a complete description of a quantum system's state. In confined systems, the wavefunction must satisfy boundary conditions and be normalized so that its square integrates to one, representing the total probability of finding the particle.

Energy Eigenstates and Quantization

In a confined system, only certain discrete energy levels, called eigenstates, are allowed. The quantization arises from the boundary conditions of the system, and each energy eigenstate is characterized by a specific quantum number.

Transcript

00:01 Okay, so in this problem, we have to calculate the probability in finding a particle between x and x plus the x.

00:12 And we have, i believe, four different positions.

00:18 So actually intervals of positions.

00:21 So first of all, let's remember what is probability.

00:24 So let's put in here in black, actually.

00:27 The probability of finding a particle between the position.

00:34 X and x plus dx is described by the integral between x and x plus d x of the conjugated wave function psi star of x that multiplies the wave function d x so we have to calculate this integral here and let's remember that for a particle in a box the wave function n of x is described by the 2 l square root of 2 l the sign of n pi x divided by l okay so since we know in this particular case that our wave function is in the first excited state that means we have n equals two so let's calculate a general solution and then we can substitute the specific values of this problem.

01:46 So first of all let's calculate this integral.

01:49 So the integral, the probability, let's call this just b, is the integral of x1 and x2, okay, so the position 1 and the position 2 of 2 divided by l, the sign square of 2 pi x divided by l d x so that's the integral we must solve first of all let's use trigonometrical identity in this sign square and we have 2 divided by l the integral between x1 and x2 of half of 1 minus the cosine of 4 pi x l d x okay therefore as we can see we have two terms and we have to calculate the probability of these two terms so first of all we can cross the half and here and to simplify the probability let's call this put in blue the general solution for the probability is going to be the 1 divided by l, which is continuing outside the integral.

03:26 The integral of the first term is a constant.

03:31 Therefore, this is just x...

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