What is the minimum possible energy in classical and quantum mechanics?

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Discussion Overview

The discussion centers around the concept of minimum possible energy in both classical and quantum mechanics, exploring whether there exists a lower limit to energy and how energy quantization applies in different contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that there is no absolute minimum energy, as the zero point of energy is arbitrary.
  • Others argue that while energy can be unbounded, specific forms of energy, such as kinetic energy, have a minimum value of zero under certain conditions.
  • A participant clarifies the inquiry to focus on minimum non-zero energy that cannot be subdivided.
  • It is suggested that energy is quantized in systems with bound states, with a quantum of energy representing the minimum in such cases.
  • Questions arise regarding the definition of "bound state" and whether there are scenarios where energy is not quantized.
  • An example is provided regarding the hydrogen atom, where the ground state energy is -13.6 eV, indicating quantization in bound states.
  • Another participant mentions that free particles can have a continuous spectrum of energy, suggesting a distinction between bound and free states.
  • Discussion includes the concept of energy bands in semiconductors, indicating continuous energy states separated by gaps.
  • There is a question about the continuity of kinetic energy in matter, leading to a comparison of classical and quantum mechanics perspectives.
  • One participant explains that in classical mechanics, kinetic energy is continuous, while in quantum mechanics, energy can be either quantized or continuous depending on the system setup.

Areas of Agreement / Disagreement

Participants express differing views on the existence of a minimum energy threshold, with some asserting that energy can be continuous while others emphasize quantization in specific contexts. The discussion remains unresolved regarding the conditions under which energy is quantized or continuous.

Contextual Notes

The discussion highlights the dependence of energy characteristics on the system's state (bound vs. free) and the definitions used, but does not resolve the implications of these distinctions.

Icebreaker
Is there an amount of energy than which there is no lower? That is, the minimum amount of energy possible.
 
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The zero of most forms of energy is an arbitrary choice, so the answer is "no". There is no minimum to energy in general.

On the other hand, every type of energy has a minimum in a particular situation. For example, the kinetic energy of an object is related to its speed. Since the speed can never be less than zero, the lowest kinetic energy an object can have is zero.
 
Oops, I meant to ask minimum non-zero energy. An amount of energy that simply can not be subdivided.
 
Icebreaker said:
Oops, I meant to ask minimum non-zero energy. An amount of energy that simply can not be subdivided.

Energy is quantised in any system where there is a bound state. That quantum of energy is indeed the minimum.
 
Please clarify "bound state". Are there instances where energy is not quantized?
 
An elemtary example of energy quantization is in the H atom, where the lowest possible (ground state) energy of the electron is about -13.6 eV (minimum 13.6 eV is needed to free it from the H atom when it's in this state). And as long as the electron is attached to the H atom, it can only have certain discrete values of energy, E_n = \frac{-13.6 \mbox{eV}}{n^2}, where n is called the energy level and can be any integer greater than or equal to 1.

A free particle on the other hand (for example an electron that isn't attached to a nucleus), can have any positive energy (E is a continuous spectrum), as far as I know.
 
You can also have bands of continuous energy states separated by an energy gap. Semiconductors and band insulators are such examples.

Zz.
 
So the kinetic energy of some matter is continuous?
 
Icebreaker said:
So the kinetic energy of some matter is continuous?

Which model of reality are you talking about? We have two very successful ones.

In the one called "Classical Mechanics" the kinetic energy of a body is continuous and is given by
\frac{1}{2}m|\dot{x}+\dot{y}+\dot{z}|

In the other one (which has a greater domain of applicability) called "Quantum Mechanics" the classical concept of energy we can extract from some physical situation can be either quantised, which means it's not continuous, or continuous depending on how the system is set up. We usually determine the states and their corresponding energies by solving the following equation:
\hat{H}|\psi\rangle=E|\psi\rangle

If you do not understand the symbols/ideas involved I suggest you read a book on the subject.
 

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