The energy of a linear harmonic oscillator is E=px^2 / 2 m+C...

The energy of a linear harmonic oscillator is $E=p_{x}^{2} / 2 m+C x^{2} / 2$. (a) Show, using the uncertainty relation, that this can be written as $$ E=\frac{h^{2}}{32 \pi^{2} m x^{2}}+\frac{C x^{2}}{2} $$ (b) Then show that the minimum energy of the oscillator is $h v / 2$ where $$ v=\frac{1}{2 \pi} \sqrt{\frac{C}{m}} $$ is the oscillatory frequency. (Hint: This result depends on the $\Delta x \Delta p_{x}$ product achieving its limiting value $\hbar / 2$. Find $E$ in terms of $\Delta x$ or $\Delta p_{x}$ as in part (a), then minimize $E$ with respect to $\Delta x$ or $\Delta p_{x}$ in part (b). Note that classically the minimum energy would be zero.)

The energy of a linear harmonic oscillator is $E=p_{x}^{2} / 2 m+C x^{2} / 2$. (a) Show, using the uncertainty relation, that this can be written as
$$
E=\frac{h^{2}}{32 \pi^{2} m x^{2}}+\frac{C x^{2}}{2}
$$
(b) Then show that the minimum energy of the oscillator is $h v / 2$ where
$$
v=\frac{1}{2 \pi} \sqrt{\frac{C}{m}}
$$
is the oscillatory frequency. (Hint: This result depends on the $\Delta x \Delta p_{x}$ product achieving its limiting value $\hbar / 2$. Find $E$ in terms of $\Delta x$ or $\Delta p_{x}$ as in part (a), then minimize $E$ with
respect to $\Delta x$ or $\Delta p_{x}$ in part (b). Note that classically the minimum energy would be zero.)

Key Concepts

Heisenberg Uncertainty Principle

This principle states that it is impossible to simultaneously determine the exact position and momentum of a particle, and that their uncertainties are bounded by a minimum value (specifically, ?x ?p ? ?/2). In the context of the harmonic oscillator, this principle is used to relate the uncertainty in momentum to the uncertainty in position, ensuring that even the lowest energy state (the ground state) has a non-zero energy.

Quantum Harmonic Oscillator

The quantum harmonic oscillator is a fundamental model in quantum mechanics that describes a particle subject to a restoring force proportional to its displacement (resulting in a quadratic potential). Unlike its classical counterpart, the quantum oscillator exhibits quantized energy levels, meaning that the energy can only take on discrete values, with the ground state energy being non-zero.

Zero-Point Energy

Zero-point energy is the residual energy that remains in a quantum system even at its lowest possible energy state. For a quantum harmonic oscillator, this energy arises due to the restrictions imposed by the Heisenberg uncertainty principle, leading to a minimum energy of h?/2. This concept highlights the intrinsic quantum fluctuations that prevent a system from ever being completely at rest.

Energy Minimization via Variational Methods

In the analysis of the quantum harmonic oscillator, one often employs a variational approach where the energy is expressed in terms of an uncertainty parameter (such as ?x) and then minimized with respect to that parameter. This method allows one to derive the lowest possible energy state by optimizing the balance between kinetic and potential energy contributions subject to the constraints imposed by the uncertainty principle.

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