Ch 40: Quantum Mechanics I: Wave Functions

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 40: Quantum Mechanics I: Wave Functions

Problem 39

Chapter 40, Problem 39

For the ground level of a harmonic oscillator, $increment x increment p sub x equals h with stroke slash 2$ . Do a similar analysis for an excited level that has quantum number $n$ . How does the uncertainty product $∆ x ∆ p_{x}$ depend on $n$ ?

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Start by recalling the uncertainty principle: $ \Delta x \Delta p_x \geq \frac{\hbar}{2} $. For the ground state of a quantum harmonic oscillator, the uncertainty product $ \Delta x \Delta p_x $ is exactly $ \frac{\hbar}{2} $. For excited states, we need to analyze how the uncertainties in position and momentum change with the quantum number $ n $.

The wavefunctions of a quantum harmonic oscillator are described by Hermite polynomials multiplied by a Gaussian envelope. The quantum number $ n $ determines the energy level $ E_n = \left(n + \frac{1}{2}\right) \hbar \omega $, where $ \omega $ is the angular frequency of the oscillator. As $ n $ increases, the wavefunction spreads out, leading to larger uncertainties in position $ \Delta x $.

The position uncertainty $ \Delta x $ can be estimated from the spatial extent of the wavefunction. For higher $ n $, the wavefunction's spread increases approximately as $ \sqrt{n} $, so $ \Delta x \propto \sqrt{n} $.

The momentum uncertainty $ \Delta p_x $ is related to the position uncertainty by the oscillator's energy. Since $ E_n = \frac{1}{2} m \omega^2 (\Delta x)^2 + \frac{1}{2} \frac{(\Delta p_x)^2}{m} $, and $ E_n \propto n $, it follows that $ \Delta p_x \propto \sqrt{n} $ as well.

Combining the dependencies of $ \Delta x $ and $ \Delta p_x $ on $ n $, the uncertainty product becomes $ \Delta x \Delta p_x \propto n $. Thus, the uncertainty product increases linearly with the quantum number $ n $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Harmonic Oscillator

A harmonic oscillator is a system that experiences a restoring force proportional to the displacement from its equilibrium position. In quantum mechanics, it is described by quantized energy levels, where the ground state corresponds to the lowest energy level. The behavior of a harmonic oscillator is fundamental in understanding various physical systems, including molecular vibrations and quantum fields.

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that certain pairs of physical properties, like position (∆x) and momentum (∆p_x), cannot be simultaneously measured with arbitrary precision. Specifically, the product of the uncertainties in these measurements is bounded by ħ/2, where ħ is the reduced Planck's constant. This principle highlights the intrinsic limitations of measurement in quantum mechanics and is crucial for understanding the behavior of quantum systems.

Quantum Number n

The quantum number n is a non-negative integer that quantizes the energy levels of a quantum system, such as a harmonic oscillator. Each value of n corresponds to a specific energy level, with higher values indicating higher energy states. The dependence of the uncertainty product ∆x∆p_x on n reflects how the spatial and momentum uncertainties change as the system transitions between different energy levels, illustrating the wave-particle duality of quantum mechanics.

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Textbook Question

While undergoing a transition from the $n equals 1$ to the $n equals 2$ energy level, a harmonic oscillator absorbs a photon of wavelength $6.50$ $μ$ m. What is the wavelength of the absorbed photon when this oscillator undergoes a transition (a) from the $n equals 2$ to the $n equals 3$ energy level and (b) from the $n equals 1$ to the $n equals 3$ energy level?

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For the ground level of a harmonic oscillator, ∆x∆px=ħ/2∆x∆p_x = ħ/2. Do a similar analysis for an excited level that has quantum number nn. How does the uncer­tainty product ∆x∆px∆x∆p_x depend on nn?

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Key Concepts

Harmonic Oscillator

Heisenberg Uncertainty Principle

Quantum Number n

For the ground level of a harmonic oscillator, $increment x increment p sub x equals h with stroke slash 2$ . Do a similar analysis for an excited level that has quantum number $n$ . How does the uncertainty product $∆ x ∆ p_{x}$ depend on $n$ ?