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

[Icebreaker]

Is there an amount of energy than which there is no lower? That is, the minimum amount of energy possible.

 

[James R]

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.

 

[Icebreaker]

Oops, I meant to ask minimum non-zero energy. An amount of energy that simply can not be subdivided.

 

[masudr]

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.

 

[Icebreaker]

Please clarify "bound state". Are there instances where energy is not quantized?

 

[rune]

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.

 

[ZapperZ]

You can also have bands of continuous energy states separated by an energy gap. Semiconductors and band insulators are such examples.

Zz.

 

[Icebreaker]

So the kinetic energy of some matter is continuous?

 

[masudr]

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.