Question

Zero point energy is a direct consequence of
• Pauli's exclusion principle
• Heisenberg uncertainty principle
• Conservation of energy
• The Schr\"odinger equation's linearity

• A. Pauli's exclusion principle
• B. Heisenberg uncertainty principle
• C. Conservation of energy
• D. The Schr\"odinger equation's linearity

Hint:

A quantum particle can never have: \[ x=0 \text{ and } p=0 \] simultaneously. Therefore quantum systems always possess minimum energy called zero point energy.

Answer 1

The Correct Option is B

Concept: Zero point energy is the minimum possible energy possessed by a quantum system even at absolute zero temperature. Unlike classical physics, quantum mechanics predicts that: \[ \boxed{ \text{A particle can never be completely at rest} } \] This phenomenon arises fundamentally from the Heisenberg uncertainty principle.

Step 1: Understand zero point energy. For systems such as:
• Harmonic oscillator
• Particle in a box the lowest energy is not zero. For example, in a harmonic oscillator: :contentReference[oaicite:0]{index=0} This finite minimum energy is called: \[ \boxed{ \text{Zero point energy} } \]

Step 2: Apply Heisenberg uncertainty principle. Heisenberg uncertainty principle states: :contentReference[oaicite:1]{index=1} This means:
• Position and momentum cannot both be known exactly. If a particle had zero energy: \[ p=0 \] and it would also need exact position. This would violate: \[ \boxed{ \text{Uncertainty principle} } \] Hence a particle must always possess some minimum kinetic energy.

Step 3: Analyze Option 1. Pauli exclusion principle applies to:
• Fermions
• Electron arrangements It is unrelated to zero point energy. Hence incorrect.

Step 4: Analyze Option 2. Heisenberg uncertainty principle directly explains why a particle cannot have: \[ x=0,\quad p=0 \] simultaneously. Thus zero point energy is a consequence of: \[ \boxed{ \text{Heisenberg uncertainty principle} } \] Hence Option 2 is correct.

Step 5: Analyze remaining options.
• Conservation of energy does not explain zero point energy.
• Linearity of Schr\"odinger equation also does not produce zero point energy directly. Thus Options 3 and 4 are incorrect.

Step 6: Choose the correct answer. Therefore: \[ \boxed{ \text{Zero point energy is due to Heisenberg uncertainty principle} } \] Hence the correct option is: \[ \boxed{(2)} \] Final Conclusion: Zero point energy originates from: \[ \boxed{ \text{Heisenberg uncertainty principle} } \] Hence, the correct answer is: \[ \boxed{(2)} \]