91Ó°ÊÓ

The Schrödinger equation for the quantum- mechanical harmonic oscillator in Eq. (40.44) may be written as $$ -\frac{\hbar^{2}}{2 m} \psi^{\prime \prime}+\frac{1}{2} m \omega^{2} x^{2} \psi=E \psi $$ where \(\omega=\sqrt{k^{\prime} / m}\) is the angular frequency for classical oscillations. Its solutions may be cleverly addressed using the following observations: The difference in applying two operations \(A\) and \(B,\) in opposite order, called a commutator, is denoted by \([A, B]=A B-B A\). For example, if \(A\) represents differentiation, so that \(A f=\frac{d f}{d x},\) where \(f=f(x)\) is a function, and \(B\) represents multiplication by \(x,\) then \([A, B]=\left[\frac{d}{d x}, x\right] f=\frac{d}{d x}(x f)-x \frac{d f}{d x} .\) Applying the product rule for differentiation on the first term, we determine \(\left[\frac{d}{d x}, x\right] f=f,\) which we summarize by writing \(\left[\frac{d}{d x}, x\right]=1 .\) Similarly, consider the two operators \(a_{+}\) and \(a_{-}\) defined by \(a_{\pm}=1 / \sqrt{2 m \hbar \omega}\left(\mp \hbar \frac{d}{d x}+m \omega x\right)\) (a) Determine the commutator \(\left[a_{-}, a_{+}\right]\). (b) Compute \(a_{+} a_{-\psi}\) in terms of \(\psi\) and \(\psi^{\prime \prime},\) where \(\psi=\psi(x)\) is a wave function. Be mindful that \(\frac{d}{d x}(x \psi)=\psi+x \psi^{\prime} .\) (c) The Schrödinger equation above may be written as \(H \psi=E \psi,\) where \(H\) is a differential operator. Use your previous result to write \(H\) in terms of \(a_{+}\) and \(a_{-}\) (d) Determine the commutators \(\left[H, a_{\pm}\right] .\) (e) Consider a wave function \(\psi_{n}\) with definite energy \(E_{n},\) whereby \(H \psi_{n}=E_{n} \psi_{n} .\) If we define a new function \(\psi_{n+1}=a_{+} \psi_{n},\) then we can readily determine its energy by computing \(H \psi_{n+1}=H a_{+} \psi_{n}=\left(a_{+} H+\left[H, a_{+}\right]\right) \psi_{n}\) and comparing the result with \(E_{n+1} \psi_{n+1}\). Finish this calculation by inserting what we have determined to find \(E_{n+1}\) in terms of \(E_{n}\) and \(n .\) (f) We can prove that the lowest-energy wave function \(\psi_{0}\) is annihilated by \(a_{-},\) meaning that \(a_{-} \psi_{0}=0 .\) Using your expression for \(H\) in terms of \(a_{\pm},\) determine the ground-state energy \(E_{0}\). (g) By applying \(a_{+}\) repeatedly, we can generate all higher states. The energy of the \(n\) th excited state can be determined by considering \(\mathrm{Ha}_{+}^{n} \psi_{0}=E_{0} \psi_{0}\). Using the above results, determine \(E_{n}\). (Note: We did not have to solve any differential equations or take any derivatives to determine the spectrum of this theory.)

Step by step solution

01

Determining the commutator [\(a_{-},a_{+}\)]

To find the commutator, we use the commutation relation [\(A,B\)] = \(AB - BA\). Hence, the commutator [\(a_{-},a_{+}\)] will be \(a_{-}a_{+} - a_{+}a_{-}\). After the calculations, the commutator will yield a result of 1.

02

Compute \(a_{+} a_{-\psi}\) in terms of \(\psi\) and \(\psi^{\prime \prime}\)

Substitute the given functions of \(a_{+}\) and \(a_{-}\) into the equation \(a_{+} a_{-\psi}\), simplify, and express it in terms of \(\psi\) and \(\psi^{\prime \prime}\). After simplification, you end up with the expression: \(-\frac{1}{2m}\psi^{\prime\prime} + \frac{1}{2}m\omega^2x^2\psi\).

03

Rewriting the Schrödinger equation using the operators \(a_{+}\) and \(a_{-}\)

Taking the result from Step 2, you can observe that the right-hand side of Schrödinger equation is equivalent to \(a_{+} a_{-\psi}\). Therefore, the differential operator \(H\) can be written as \(a_{+} a_{-}\) in the Schrödinger equation \(H\psi = E\psi\).

04

Determining the commutators [\(H, a_{\pm}\)]

For this step you apply the calculation for a commutator as in Step 1, except that now you are doing it for the commutators [\(H, a_{\pm}\)], which evaluates to be \(\pm \hbar \omega a_{\pm}\).

05

Determining \(E_{n+1}\) in terms of \(E_{n}\) and \(n\)

We use the function \(\psi_{n+1} = a_{+}\psi_{n}\) and the commutator [\(H, a_{+}\)] to compute \(H\psi_{n+1}\) which allows us to find \(E_{n+1}\) in terms of \(E_n\). This calculation finds that \(E_{n+1} = (n+1) \hbar \omega\).

06

Determining the ground-state energy \(E_{0}\)

Given that the lower-energy wave function \(\psi_0\) is annihilated by \(a_{-}\), \(a_{-} \psi_0 = 0\), and using the formula for \(H\) in terms of \(a_{+}\) and \(a_{-}\), we can determine \(E_0 = \frac{1}{2} \hbar \omega\).

07

Determining \(E_n\)

Repeated application of \(a_{+}\) allows us to generate all higher states. The energy of the \(n\)th excited state can be determined by considering \(Ha_{+}^n\psi_{0} = E_{0}\psi_{0}\), and show that it is \(E_{n} = \left(n + \frac{1}{2}\right)\hbar \omega\). Thus the spectrum of the harmonic oscillator is effectively determined without solving any differential equations or taking any derivatives.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Schrödinger Equation

The Schrödinger Equation is a fundamental equation in quantum mechanics. It provides a way to calculate how quantum states evolve over time. This equation, typically expressed as \( H \psi = E \psi \), is essential for describing how the wave function, \( \psi \), moves and changes. In the context of a quantum harmonic oscillator, the Schrödinger equation describes the quantized nature of the energy levels of the system.
The equation takes on a specific form for a quantum harmonic oscillator, involving variables like mass \( m \), angular frequency \( \omega \), and Planck's constant \( \hbar \). It can be written as:

• \( -\frac{\hbar^{2}}{2 m} \psi^{\prime \prime} + \frac{1}{2} m \omega^{2} x^{2} \psi = E \psi \)

This equation accounts for both the kinetic and potential energy of the system. The solutions to this equation give rise to discrete energy levels, explaining phenomena such as the quantization of energy.

Commutator

In the realm of quantum mechanics, a commutator offers insight into the compatibility of two operations. It is denoted by \([A, B] = AB - BA\). The commutator measures the extent to which two operations fail to commute, meaning how their order of application influences the result.
In the given exercise, commutators play a key role in analyzing operators involved in the harmonic oscillator problem. For instance, for the operators \(a_{+}\) and \(a_{-}\), the commutator is given by \([a_{-}, a_{+}] = a_{-}a_{+} - a_{+}a_{-}\). The result of this commutator turns out to be 1, which is crucial for the algebraic handling of the quantum harmonic oscillator.
Understanding commutators helps in constructing mathematical expressions without direct computation of differential equations, simplifying complex problems into algebraic manipulations.

Ladder Operators

Ladder operators are a mathematical tool used in quantum mechanics, particularly advantageous for systems like the quantum harmonic oscillator. These operators are symbols like \(a_{+}\) and \(a_{-}\), which help "step" through the energy states of a quantum system.

• \(a_{+}\) is called the "raising" or "creation" operator.
• \(a_{-}\) is referred to as the "lowering" or "annihilation" operator.

Both operators work together, allowing movement between quantized energy levels. For example, applying \(a_{+}\) to a state \(\psi_{n}\) increases the energy, moving to \(\psi_{n+1}\). Conversely, using \(a_{-}\) decreases the energy, stepping down to a lower state.
These operators are defined in the context of the harmonic oscillator as follows:\[ a_{\pm} = \frac{1}{\sqrt{2m \hbar \omega}} \left( \mp \hbar \frac{d}{dx} + m \omega x \right) \] These expressions involve derivatives and position components, connecting quantum mechanics' probabilistic and wave-like nature with physical quantities of the harmonic oscillator.

Quantum Mechanics

Quantum Mechanics is the branch of physics that deals with the behavior of small particles, often at the atomic or subatomic level. Unlike classical mechanics, it embraces the concept of wave-particle duality, suggesting that particles could be wave-like and vice versa. This field relies on probabilities to depict the position and momentum of particles.
In quantum mechanics, operators and commutators represent physical observables and encapsulate the probabilistic nature of measurement. Systems like the quantum harmonic oscillator are studied to demonstrate essential quantum concepts such as discrete energy levels and the unpredictability of measurement outcomes.
The harmonic oscillator model, a cornerstone of quantum mechanics, illustrates how particles exhibit wavelike properties and how their energies are quantized—limited to specific values. This model can describe various systems, from vibrational modes in molecules to more complex fields in physics, highlighting the versatility and depth of quantum mechanics applications.