An electron is confined to a one-dimensional region in which...

An electron is confined to a one-dimensional region in which its ground-state $(n=1)$ energy is 2.00 eV. (a) What is the length $L$ of the region? (b) What energy input is required to promote the electron to its first excited state?

Key Concepts

Quantization of Energy Levels

In confined quantum systems, allowed energy values occur at specific discrete levels rather than a continuous range. These energy levels depend on the square of an integer quantum number and are inversely proportional to the square of the length of the confinement region. This inherent quantization explains the stepwise energy spectrum observed in various quantum systems.

Energy Transitions

When a quantum particle moves between different energy states, the energy absorbed or emitted equals the difference between these quantized levels. This concept of energy transitions is crucial for understanding processes such as excitation, absorption, and emission of radiation in quantum systems.

Particle in a Box Model

This is a foundational quantum mechanics model where a particle is confined within an infinitely deep potential well with fixed boundaries. The model demonstrates that the particle cannot possess arbitrary energies; instead, it has discrete, quantized energy levels determined by the size of the confinement and the quantum number. It serves as a simple illustration of how boundary conditions lead to energy quantization.

Transcript

00:01 So we have the first energy state.

00:04 We have the ground state energy, essentially, of two electron volts.

00:09 And we're going to set this equal to the equation for a particle in a box, which is planck's constant over eight times the mass of the particle, times the length squared of the box.

00:27 I left a space here because we also would like to know the energy in joules, because we're going to be working in units where the energy needs to be in joules.

00:39 And you would simply multiply two electron volts by 1 .6 times 10 to the minus 19.

00:52 And this would give you the energy in joules.

00:58 So that is 3 .6 o times 10 to the negative 19.

01:08 So we'll place that value right here.

01:17 And from this equation, which i'll just highlight, we can obtain the length l, and we're going to end up taking a square root because we have l squared.

01:29 So we'll place this equation down here and just work to get l.

01:52 So the first thing we want to do is multiply both sides by l squared, and then we're going to divide also by 3 .60 times 10 to the minus 19.

02:20 And of course right here, the length squared will cancel.

02:26 And now we have, and also this number will cancel on the left.

02:31 So we have l squared is equal to h squared over 8 times m times the, what's really the energy, 3 .60 times 10 to the minus 19.

02:45 And we just now have to substitute in the mass and planx constant.

02:55 And so we're dealing with an electron, so we'll place the mass right here.

03:16 And in the numerator, we have the square of planck's constant.

03:20 So we have 6 .626 times 10 to the minus 34 squared.

03:29 And when you calculate this, you get, well, actually, we need the square root of it.

03:39 So let me go ahead and rewrite this a certain way.

03:44 Just place a square root over everything, and the length will equal 4 .34 times 10 to the minus 10 meters...

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