An electron confined to a one-dimensional box has energy levels given by the equation: En=n^2 h^

An electron confined to a one-dimensional box has energy...

Key Concepts

Quantum Confinement in One-Dimensional Systems

This concept refers to the restriction of a particle's motion to a single spatial dimension, often modeled as a particle in a box. When a particle such as an electron is confined in a narrow region, its energy levels become discrete rather than continuous. This is a foundational idea in quantum mechanics that illustrates how spatial confinement leads to quantization of energy.

Energy Quantization in a Particle-in-a-Box

In the particle-in-a-box model, the allowed energy states of a confined particle are determined by a quantization condition that depends on the quantum number n, the mass of the particle, and the size of the confining box. The energy levels are given by an equation that shows the quadratic dependence on the quantum number, emphasizing that higher quantum states have significantly greater energy, which is a direct consequence of the boundary conditions imposed by the confined space.

Energy Transitions and Photon Absorption

When a quantum system transitions between energy levels, it absorbs or emits energy in the form of photons. The energy difference between the initial and final states is directly related to the frequency (or wavelength) of the absorbed or emitted photon. This principle is central to understanding spectroscopic transitions and the quantized interaction between matter and electromagnetic radiation.

Relationship between Energy Difference and Wavelength

The connection between the energy of a photon and its wavelength is quantified by the equation E = hc/?, where h is Planck's constant and c is the speed of light. This relationship allows one to calculate the wavelength of light required to induce a transition between quantum states by equating the energy difference between those states to the energy of the photon. This key concept bridges the concepts of quantum mechanics with experimental spectroscopy.

Transcript

00:01 Hey, hello people, how's your going? for this question, we have to calculate the energy available in n equals 1, 2, and 3, respectively in the box with length of 155 p .m.

00:13 So first, let's find the equation that we're going to use.

00:16 The equation that we're going to use is as follows.

00:19 The energy available at n -level energy is n -square.

00:25 The energy level square times each square, which is the planx constant square, divided by 8 times the mass, times the length, the box.

00:35 Here's your energy equation, and when we need to calculate n equals 1, we just have to plug this number in.

00:42 Very simple, we have en equals to 1 square times the planx constant square, which is 6 .63, times 10 to the power of negative 34, joules times second, second square and all this divided by 8 times the mass of the electron which is 9 .1 times 10 to the power of negative 33 kilograms times the length of the box which is 155 p .m.

01:10 Do a quick unit conversion 10 to the power of 12 p .m.

01:13 Is the same as 1 meter and we have 2 squaredness and we're going to get an answer of 2 .51 times 10 to the power of negative 18 joules.

01:26 4n equals to 2, the en calculation is exactly the same, except for that the n squared this time is going to be another new n, which is 2 here.

01:37 So it's going to be 2 squared times each square.

01:40 Everything else will remain the same, and we're going to get 1 .00 times 10 to the power of negative 17 joules.

01:47 And for n equals to 3, we're going to use 3 square, or 9 times h, and we're going to get 2 .26 times 10 to the power.

01:56 Power of negative 70 joules.

02:00 That's the first part of the question.

02:02 Now for the second part, we have to calculate the weight length of the light required for transitions from n equals to 1 all the way to n equals to 2, and from n equals 2 to n equals to 3.

02:14 Now, if we want to calculate the wavelength of the light, you need to note the energy required for the transition.

02:20 If we want to know the energy required for the transition, we have to know the energy before the electron jump and energy after the electron jump, and we can calculate the difference, so to speak, the transition energy.

02:35 Let's calculate the transition energy.

02:37 The energy difference is the final n level after the jump square times h squared.

02:45 This is exactly almost the same equation as before, and we're subtracting by the initial energy level, which is initial n square times h squared divided by 8m l squared.

03:01 We note that there is a lot of similarities between those two expressions, so we can combine them together.

03:07 We have h2 over 8ml square times by the final n level minus the initial n level square, all of them.

03:16 And now let's rearrange the delta e .n or the changing energy into something more familiar or something that contains the lambda, such as the hc over lambda, right? this is what we familiar with.

03:30 We need to find the wavelength, so that's why we're expressing it in terms of the lambda.

03:35 Everything on the right will remain the same as before, minus n -i -square bracket.

03:44 Now we know that there are two hs on both sides, so we can cancel one of them out, leaving only one h on the right.

03:50 And at the same time, let's do another rearrangement...

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