Correct option-4

Concept:

• Quantum mechanics:It is a branch of mechanics deals with the study ofmotion and interactionofsubatomic particleslike theelectron, proton, neutrons,etc. with the help of mathematical descriptions (linear algebra, vector space, differential equations, special integrations, etc.)
• InQuantum mechanicsthere are somewave quantitiesassociated withquantum particlesorsubatomic particles.
• The most popular symbol is used to represent the wave quantity associated with thequantum particles is\(\psi (r,t)\)whatwe call the wave function of the particle.
• The wave function\(\psi (r,t)\)represents thestateof a quantum particle in quantum mechanics at a given instant of time.
• The stateof a Quantum particle in quantum mechanics at any instant of time is described by two quantities, position and velocity.

Calculation:

In quantum mechanics, a quantum particle in a given system is treated as a wave-particle.

Actually, they are assumed as wave packets which is a linear combination of several waves that interfere constructively.

The energy associated with such wave-particle is given by-

\(E=h\vartheta \) ----(i)

Since\(\vartheta =\frac{ω }{2\pi }\) ----(ii)

where,

ω is the angular frequency

On substituting the given value of\(\vartheta\)from equation (ii) in equation (i), we get

\(E=h\frac{ω }{2\pi }=\hbar ω \) Here\(\hbar =\frac{h}{2\pi }\)

\(\therefore E=\hbar ω \)

Thus, option-4is the correct answer.