For a particle exhibiting simple harmonic motion in 1-dimension, the uncertainty in its position in the state having the following schematic wave function is (zero point energy, Eo = )

Concept:

The Simple Harmonic Oscillator (SHO) in one dimension is a fundamental concept in physics and quantum mechanics. It describes the motion of a particle that is subjected to a restoring force proportional to its displacement from a fixed equilibrium position. In a one-dimensional SHO, the particle can move only along a straight line.

Explanation:

According to uncertainty principle

.      -----(1)

For SHO <x> = 0

→ Calculating value of <x>²

V = 1/2 kx²

.       ---(2)

According to Viral Theorem of SHO,

Where,

<T> = Average Kinetic energy

<V> = Average Potential energy

Therefore, 

or, 

According to Vibrational quantum,

Where, n = no. of nodes 

.^.. 

or, E_n = 7E_o

Putting the value of En in eq(2)

.^.. 

Now, putting the value of<x²> in eq (1)

Conclusion: 

Therefore, the uncertainty of the particle in 1-D SHO is