Is it right that the given k in excercise 3.1 of the Einstein Model is the spring constant instead of the entire returning force?
The answer model sugguests this by using k as wave constant. However, in the question k is introduced as the returning force.
If I am well, k should be the wave constant. Otherwise some information is missing as the amplitude of the molecules cannot be known by the given information only.
If I am right, this is a minor error. However, it costs students like me much time by trying to solve this.
You are definitely correct, the returning force would be -k\bm{r}; k by itself would be the spring constant. “Spring constant”, however wouldn’t be accurate neither because we’re talking about a 3D harmonic potential, and so the precise term would be “harmonic force constant” or just “force constant”. Do you think if we wrote "the same harmonic force constant k" the problem formulation would be clearer?
As a broader advice (since you wrote that interpreting such formulation costs you time): try to focus on the physical situation that is described: physics problems aren’t so much about following instructions, as they are about understanding a physical situation.
Let me illustrate how it applies to the problem 3. The atoms interact via electromagnetic forces, and these forces only depend on the atomic charges and the numbers of electrons. These forces don’t depend on the number of neutrons in an atom—the property that distinguishes different isotopes. Additionally, in the context of analyzing thermal properties of a material, we don’t consider atoms at a fixed position, and therefore when we speak about the force strength it wouldn’t make sense to compare the values of force acting on an atom at a fixed moment in time. This is why interpreting the problem formulation as being about the elastic constant is the only meaningful way.
In other words, when you read a problem formulation rather than thinking “what are the exact definitions of the quantities involved” consider “what is the physical question that we are trying to understand”. This approach requires a very different attitude, and that it is an absolutely crucial state of mind in analyzing complex problems. The overall model of the problem is what you will always need to interpret, while the exact definitions of different parts will be of little help.
Thank you for your reply and your explanation and help. Your explanation makes that I understand it better. I indeed saw a relation like in a spring, but now I understand that a spring constant would not be accurate to a 3D harmonic potential and that a elastic constant is a better parallel.
Yes, I think this formulation would be clearer. Maybe some students have to think what this constant means. They have to realise that in a harmonic potential the returning force must be linear. Maybe some students will fail to realise this and need help. If you do not want that to happen, you could explain this in the question. However, I think this would also be a good excercise for those students.
Thank you for your broader advice. I will try to focus on the physical situation that is described, less on the exact definition. Actually I got in this question soon the idea that the question would far more easier if k would be a constant. Maybe I should have listened to that earlier. However, I think it is for students like me not a good attitude to change the question if I see no solution. Mostly the reason why I cannot solve the question, is because I make an error or do not understand the needed physics well. In the end the question is made by experts like you. If you state something what sounds to me as physically wrong, usually I am wrong. In those situations, mostly I do not understand the situation well. This is especially the case in questions like this in which the authors have the time to remove errors. That makes that I try a lot to solve the situation with the given information.
If this happens in the exercises, that is unfortunately to students like me. In the end we can ask about the question on this forum and in the working classes. However, I hope it will not happen in the tests and exam. Because students like me will than spend lots of expensive time trying to solve the question like how it is posted (and maybe fail to do that because it is an impossible question), while you mean a different question.
Ps. Of course, in some situations it is easier for me to see what is really meant in the question. If you give in a question for example the amount of energy in terms of Hertz, I will understand that that means a certain energy in joules. I will also understand that it has in this course often no special use to state the energy in terms of joules. This understanding is partly because you switch often in this course between those quantities to say the same. Therefore, I understand what you mean. In this question, that was not the case to my opinion.