Prove zero point energy without calculation?

In summary, to prove that a quantum mechanical system with bounded particles has a non-zero ground state energy, one can use the Heisenberg principle and the concept of uncertainty in momentum to derive the expectation value of the particle's momentum. Another approach would be to take a Taylor series around the global minimum and consider the example of a harmonic oscillator. However, a simpler conceptual answer would be to acknowledge that even the lowest vibrational state of a system must have some energy, on the order of ~\hbar.
  • #1
Gerenuk
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What is an elegant way to prove that quantum mechanical system (with bounded particles?) have some non-zero ground state energy?
 
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  • #2
How does one prove that the result of a calculation is non-zero (or non-any value) without doing a calculation?
 
  • #3
Without the full calculation of energy. Basically the easiest way possible, whatever that is.
 
  • #4
Gerenuk said:
Without the full calculation of energy. Basically the easiest way possible, whatever that is.

Well, should you take Heisenberg principle as true, there is no much caliculation involved : as an example, take a free particle and localize it ... SNIP ...such uncertainty in momentum, it's easy to derive p2 expectation value thus kinetic energy.

erorr : I didn't read the question as careful as I thought I did =) Having general QM problem in mind, I am no longer sure ... maybe taking Taylor series around global minimum and recalling oscillator, but that is hardly "without of caliculation".
 
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  • #5
A simple hand-waving conceptual answer would be something like: Consider a simple vibrating system, e.g. a harmonic oscillator. The lowest possible energy (classically) would be for the thing to simply not vibrate. Quantum mechanically, it can't do that since it'd mean having a well-defined position and momentum, in violation of the uncertainty principle.

So the lowest vibrational state must have some energy, on the order of ~[tex]\hbar[/tex].
 

Related to Prove zero point energy without calculation?

1. What is zero point energy?

Zero point energy refers to the lowest possible energy that a quantum mechanical physical system may have. It is the energy that particles possess even at absolute zero temperature, when all other forms of energy have been removed.

2. How is zero point energy related to the uncertainty principle?

The uncertainty principle states that there is a limit to how precisely we can know the position and momentum of a particle. This uncertainty leads to the existence of zero point energy, as even at the lowest possible temperature, particles are still in constant motion due to this uncertainty.

3. Can zero point energy be measured?

No, zero point energy cannot be directly measured as it is a theoretical concept. However, its effects can be observed through phenomena such as the Casimir effect and the Lamb shift.

4. How is zero point energy relevant to everyday life?

While zero point energy may seem like a purely theoretical concept, it has important implications in fields such as quantum mechanics and cosmology. It also provides a basis for some alternative energy theories, although currently there is no practical way to harness zero point energy.

5. How can zero point energy be proven without calculation?

While it is not possible to prove zero point energy without some level of mathematical calculation, its existence can be inferred through various physical phenomena and observations. These include the Casimir effect, the Lamb shift, and the existence of virtual particles in the quantum vacuum.

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