For a quantum particle of mass m in the ground state of a...

For a quantum particle of mass $m$ in the ground state of a square well with length $L$ and infinitely high walls, the uncertainty in position is $\Delta x \approx L .$ (a) Use the uncertainty principle to estimate the uncertainty in its momentum. (b) Because the particle stays inside the box, its average momentum must be zero. Its average squared momentum is then $\left\langle p^{2}\right\rangle \approx(\Delta p)^{2}$ . Estimate the energy of the particle. (c) State how the result of part (b) compares with the actual ground-state energy.

Key Concepts

Energy Quantization in Confined Systems

In quantum systems with boundaries, like a particle in a potential well, the particle's energy can only take specific quantized values due to the imposition of boundary conditions on its wavefunction. The ground state energy is the lowest allowed energy level, and it is directly influenced by the spatial confinement and the uncertainty in momentum.

Estimation Techniques Using the Uncertainty Principle

Using the uncertainty principle to estimate physical quantities involves approximating the minimum uncertainties in observables like momentum, based on the confinement length, and then inferring related quantities such as kinetic energy. This approach provides a way to understand the scale of quantum effects without solving the full wave equation.

Heisenberg Uncertainty Principle

This principle states that certain pairs of physical properties, such as position and momentum, have a fundamental limit to the precision with which they can be known simultaneously. It provides an inequality that relates the uncertainties in position and momentum, and is used to estimate the magnitude of one when the other is specified.

Quantum Confinement

When a particle is restricted to a finite region of space, such as inside a potential well, its allowed energy states become discrete instead of continuous. This confinement alters the physical properties of the particle and is a key concept in understanding how spatial restrictions lead to quantization of momentum and energy.

Transcript

00:01 The uncertainty in the position is approximately equal to the length, and we know that the average momentum is zero.

00:08 And this gives us the result that the square of the uncertainty in momentum is equal to the average of the momentum squared, based on how one defines uncertainty in quantum mechanics.

00:21 So we're going to use the heisenberg uncertainty principle to get an estimate of the energy of the particle.

00:29 The uncertainty principle is given as the product of the uncertainty in position and uncertainty momentum, and it's an inequality.

00:38 Their product is greater than or equal to h bar over 2.

00:42 Now, h bar is equal to plonks constant h divided by 2 pi, and it's important, it is very important to distinguish between h bar and h.

00:52 Because obviously you have this factor of 2 pi.

00:57 And so proceeding, we're going to go ahead and write the energy for a particle in a box as p squared over 2m.

01:08 And in order to approximate the energy, we can go ahead and talk about the average energy, and that we just put in brackets.

01:18 Now, these brackets that we use for averages really refer to expectations.

01:23 Values as defined in quantum mechanics.

01:26 And real quick, if you have an operator m, an expectation value for an operator is simply the integral over an interval, which in general might be from minus infinity to positive infinity, of the product of the wave functions complex conjugate with the operator and the wave function on the right.

01:52 So that's how you get an expectation value.

01:55 You can also define them over other intervals, but in general it's from minus infinity to positive infinity.

02:01 Having said that, the estimate that we're going to get for the energy is just going to be this average, and we're going to say that it's approximately equal to this, because after all, we're not working with anything exact here.

02:16 So making note of the observation up here that based on the average momentum being zero, we have this result for the screen...

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