Quantum Harmonic Oscillator: Energy Minimum from Uncertainty Principle

swarna

what is zero point energy??

tanzeel_siddiqui

In physics, the zero-point energy is the lowest possible energy that a quantum mechanical physical system may possess and is the energy of the ground state of the system. The concept was first proposed by Albert Einstein and Otto Stern in 1913. The term "zero-point energy" is a translation of the German Nullpunktsenergie. All quantum mechanical systems have a zero point energy.

chemicaltester

http://en.wikipedia.org/wiki/Zero_point_energy

here it goes
In physics, the zero point energy is the lowest possible energy that a quantum mechanical physical system may possess; it is the energy of the ground state of the system. The term "zero point" refers to the observed fact that vacuum fluctuations persist at zero degrees Kelvin.

edison

It is the energy at absolute zero.
It is the energy (non -zero ofcourse) that corresponds to lowest vibrational energy level at absolute zero for an atom, electron or particle.

edison

Quantum Harmonic Oscillator: Energy Minimum from Uncertainty Principle

The ground state energy for the quantum harmonic oscillator can be shown to be the minimum energy allowed by the uncertainty principle.

The energy of the quantum harmonic oscillator must be at least

Taking the lower limit from the uncertainty principle

Then the energy expressed in terms of the position uncertainty can be written

Minimizing this energy by taking the derivative with respect to the position energy and setting it equal to zero gives

S

olving for the position uncertainty gives

Substituting gives the minimum value of energy allowed.

This is a very significant physical result because it tells us that the energy of a system described by a harmonic oscillator potential cannot have zero energy. Physical systems such as atoms in a solid lattice or in polyatomic molecules in a gas cannot have zero energy even at absolute zero temperature. The energy of the ground vibrational state is often referred to as "zero point vibration". The zero point energy is sufficient to prevent liquid helium-4 from freezing at atmospheric pressure, no matter how low the temperature.

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Quantum Harmonic Oscillator: Energy Minimum from Uncertainty Principle