A particle of mass m is subjected to a potential V(x)={ 0, |x| a /

A particle of mass m is subjected to a potential V(x)={ ...

Key Concepts

Expectation Values

Expectation values represent the average or expected outcome of measurements for a given observable in a quantum state, calculated as the integral of the observable multiplied by the corresponding probability density. In the context of the infinite square well, typical observables include position and momentum, and their expectation values provide insights into the properties of the stationary and dynamic states.

Uncertainty Principle

The uncertainty principle states that certain pairs of observables, such as position and momentum, cannot be simultaneously measured with arbitrary precision. In practice, this is quantified by the product of the standard deviations (uncertainties) in these observables. The infinite square well serves as a platform to explore how confined states adhere to and illustrate the principles of quantum uncertainty.

Probability Densities

Probability density is defined as the square modulus of the wave function and gives the likelihood of finding a particle at a specific position in space. In quantum mechanics, visualization of probability densities over time helps in understanding the spatial distribution and dynamical behavior of quantum states, particularly when assessing the contribution of excited states and superpositions.

Eigenfunctions and Eigenvalues

Eigenfunctions are the allowed, normalized wave functions that satisfy the Schrödinger equation under the given constraints, while eigenvalues are the corresponding energy levels of the particle. In the infinite square well, these eigenfunctions take the form of sine or cosine functions that vanish at the boundaries, and the eigenvalues depend on the quantum number and the physical parameters of the system, such as mass and well width.

Stationary States and Energy Quantization

Stationary states are solutions to the time-independent Schrödinger equation that describe states of definite energy. In systems like the infinite square well, these states emerge from the quantization condition imposed by the boundary conditions, resulting in discrete energy levels. Each stationary state corresponds to a particular energy eigenvalue and its associated eigenfunction, and the quantization reflects the wave-like nature of particles in confined geometries.

Boundary Conditions

Boundary conditions are the constraints imposed on the solutions of the Schrödinger equation due to the potential's geometry and nature. In the case of the infinite square well, the wave function must be zero where the potential is infinite, which leads to specific allowable forms of the wave functions. These conditions ensure the physical acceptability of the solution, such as normalization and continuity constraints within the allowed region.

Infinite Square Well

The infinite square well is a fundamental model in quantum mechanics where a particle is confined to a finite region with infinitely high potential barriers at the boundaries. This idealized potential forces the wave function to vanish at the walls, leading to the quantization of energy and spatial confinement of the particle. The model serves as an instructive example for understanding quantization and boundary value problems in quantum mechanics.

Transcript

00:01 We start by calculating the derivatives for the wave function given as well as for the potential, and then we insert those into the schrodinger equation for the potential given.

00:14 So the first derivative of this wave function is given as negative 2ax over l squared.

00:33 We see that only the second term contributes to that derivative because the first term is a constant.

00:39 Now, we get the second derivative and we just have to that, have a constant for the second derivative, since obviously we only have a first power of x in the first derivative.

00:51 So that second derivative for the wave function is then negative 2a over l squared.

00:59 I'm going to scroll down a little bit, and we'll continue.

01:04 So when we insert this second derivative, as well as the potential into the schrodinger equation, we get the following.

01:15 We get negative h bar squared over 2m times, negative 2a over l squared as well as the potential, negative h -bar squared x squared over ml squared times l -square -minus x squared times the wave function.

01:32 And this will equal the energy times the wave function.

01:36 And there's some reducing that we can do immediately.

01:43 I'm going to go ahead and scroll down a bit more.

01:45 And we see that, for example, in the first term, the two will cancel, the twos will cancel.

01:52 Also, the minus signs will cancel because they multiply to give us positive 1.

01:59 In the second term, if we pull this l under the x squared out, we can rewrite, let me scroll up slightly here.

02:18 We can write this 1 minus x squared over l squared, having pulled the l squared out.

02:24 We can write it as l squared minus x squared, and we can write that here.

02:31 As l squared minus x squared.

02:34 This can then be written over the l squared minus x squared in the denominator, which you see right here in the denominator, and these will then cancel.

02:48 So they're brought together, and they then cancel.

02:52 The l squared that we pulled out will go here in the denominator.

02:56 It'll go next to another factor of l squared, which is already there.

03:00 But we're going to leave it in this form just for a minute.

03:03 And on the right -hand side of the side of the number, the equation right now we have the energy times the wave function which has also a factor of one minus x squared over l squared now we can also notice that the a's cancel out of all three terms so right now this simplifies greatly to h bar squared over m l squared minus h bar squared over m l squared times x squared over l squared that's the reason i've left these two l squares separate the two l squares you see were left separate because i wanted to write it in this form.

03:40 So right now we have h bar squared over ml squared in three different terms and there's a bit of factoring we can do on the left side, which we're about to do next.

03:53 You'll see that if you pull h bar square over ml squared out of both terms on the left hand side of the equation, you get this middle form right here in this middle form of the equation, and you have 1 minus x squared over l squared, which can then be set equal to the energy times 1 minus x squared over l squared.

04:17 And so we see that the energy is actually equal to h bar squared over m l squared.

04:25 And that is the energy for the wave function that we have, as well as the potential function that we have.

04:33 So now, we move on to getting a normalized wave function.

04:40 So recall that we are looking at this wave functions form over the interval that we were given as well.

04:50 And that interval once again is from negative l to l.

04:56 So when we take a normalized wave function, or we want to calculate it to be normalized, we have to set the integral up this way.

05:04 And so we go ahead and do that.

05:07 And when we square the form of the wave function, we get the following.

05:14 And so in carrying out this integral, we go ahead, we take the integral of all three terms, and we get x minus 2x cubed over 3l squared, plus x to the fifth over 5l to the fourth, and we evaluate this at minus l and l, and this is what we have so far.

05:41 So notice we've got six terms.

05:51 We've got l minus negative l, and we've got negative 2l cubed over 3l squared, minus 2l cubed over 3l squared.

06:00 And then we've also got plus l to the 5th over 5l to the 4th, l of the 5th, and 5l of the 4th again.

06:08 Now one thing you'll notice is that we've always got, well, for example, here we've got l, we've got l, but here we've got l cubed over l squared, so we're always one power of l higher in the numerator than we are in the denominator.

06:25 So that's what we would want because we want to have a dimension of length.

06:33 And that's going to be important for the normalization of this wave function.

06:39 And in doing so, you'll see that this is also going to simplify to the following.

06:44 So now in the next step, we've taken these factors, and we've simplified them to 2l minus 4l over 3 plus 2l over 5, and now we can factor the common factor of l out...

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