A certain quantum-mechanical system has the energy levels shown in the accompanying diagram. The energy levels are indexed by a single quantum number n that is an integer. (b) Which quantum numbers are involved in the transition that requires the least energy?

Question

A certain quantum-mechanical system has the energy levels shown in the accompanying diagram. The energy levels are indexed by a single quantum number n that is an integer. (b) Which quantum numbers are involved in the transition that requires the least energy?

Diagram showing energy levels indexed by quantum number n for quantum mechanics.

Accepted Answer

Welcome back everyone in this example, we need to identify the transition with the least energy based on our given diagram below. So analyzing our below diagram, we can see that we begin at the first energy level because we should recall that our end value here means that this number one is our first energy level or we're at the first shell number, so we'll write that down. So moving on, we would see that we go from the first energy level at this line here and then we get to the second energy level here and this continues up in energy all the way to the fifth energy level, where we can say that the fifth energy level is highest in energy. And so according to the prompt, we need to find the transition with the least energy. And so that means we want to move from a transition with the least magnitude. So looking at our answer choices below, we have a through D to choose from. And looking at answer choice A. It says equals to two equals four. So looking at our diagram, we can see that going from n equals two to n equals four. And we'll just represent that actually with a red line is going to give us an upward transition, which we can say is probably a pretty large and energy transition because we can see we have a pretty large area here. So let's go ahead and actually continue on considering our options before we rule out choice A But it doesn't seem like choice is a good choice moving onto choice B. It considers a transition going from the third energy level to the fifth energy level. And so looking at our diagram given we can see that going from the third energy level and will use actually the color black going to the third all the way up to the fifth. Energy level is going to give us an even higher transition in energy where we have an even taller arrow from the third to the fifth energy level. So right now let's go ahead and rule out Choice B. For sure because we know that it gives us a pretty large energy transition. Next we have choice C. Which says going from energy level one to energy level three. So drawing out that era will use to color purple on our diagram and we can see going from energy level 123 will give us this transition in purple where we have an even smaller arrow than A. And choices B. So we're going to rule out choice A. Since we can see that choice C. Gives us a smaller transition. And lastly we have choice D to consider which says going from the second energy level to the third energy level. So going back to our diagram, We will start and we'll use the color blue here going from energy level to to energy level three. And as you can see our blue line is even smaller than our purple line here. So we have a smaller energy level going from the second entry level to third energy level. And so we're gonna go ahead and rule out choice. See meaning that we can confirm that choice D. Going from the second energy level two to the third. Energy level is going to be the smallest transition in energy, and so D would be our final answer. To complete this example. I hope that everything that I reviewed was clear. But if you have any questions, please leave them down below and I will see everyone in the next practice video.

A certain quantum-mechanical system has the energy levels shown in the accompanying diagram. The energy levels are indexed by a single quantum number n that is an integer. (b) Which quantum numbers are involved in the transition that requires the least energy?

Verified Solution

Key Concepts

Quantum Numbers

Energy Transitions

Energy Level Diagram