Fundamentals of Quantum Physics 3: Quantum Harmonic Oscillator 🌚 Lecture for Sleep & Study

LECTURES FOR SLEEP & STUDY・2 minutes read

The lecture delves into the Quantum Harmonic Oscillator, focusing on solving the Schrödinger equation using ladder operators and highlighting the potential energy function's behavior. Manipulating ladder operators and the Hamiltonian is crucial for obtaining solutions to the Schrödinger equation, leading to quantization conditions and restrictions on power series structures for normalization.

Insights

The lecture focuses on the Quantum Harmonic Oscillator and ladder operators to solve the Schrödinger equation, emphasizing the potential energy function's parabolic nature.
The Hamiltonian operator's manipulation using ladder operators A+ and A- enables the rewriting of expressions, simplifying solutions to the Schrödinger equation.
The quantization of energy levels in the power series solutions for the quantum harmonic oscillator leads to specific energy values and wave function structures.
Fourier analysis plays a pivotal role in quantum mechanics, enabling the expression of wave functions in terms of position and time through integrals, showcasing differences in velocity calculations compared to classical mechanics.

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Recent questions

What is the Quantum Harmonic Oscillator?
A system described by ladder operators and Schrödinger equation.
How are wave packets created in quantum mechanics?
By summing non-normalizable states to form localized states.
What is the significance of Fourier analysis in quantum mechanics?
It involves determining wave functions through integrals.
How are Delta functions used in quantum mechanics?
To calculate expected values near specific points.
What is the relationship between energy levels and wave behavior?
Energy levels affect wave function curvature and normalization.

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Summary

00:00

Quantum Harmonic Oscillator: Solving Schrödinger Equation

The lecture focuses on the Quantum Harmonic Oscillator and solving the Schrödinger equation using ladder operators.
The potential energy of the harmonic oscillator is described as 1/2 M omega^2 x^2.
The potential energy function is graphed as a parabola, indicating the behavior of solutions to the Schrödinger equation.
Solutions to the Schrödinger equation curve downwards when energy is above potential and curve upwards when potential is above energy.
The lecture discusses the challenges of fine-tuning solutions to the Schrödinger equation.
The time-independent Schrödinger equation for the harmonic oscillator is presented.
The Hamiltonian operator is factored using ladder operators A+ and A-.
The commutator of position and momentum operators is explained.
The Hamiltonian is expressed in terms of ladder operators and constants.
The lecture explores the manipulation of ladder operators and the Hamiltonian to find solutions to the Schrödinger equation.

19:28

Hamiltonian simplifies equations, generates infinite energy states.

The Hamiltonian is crucial for applying the Schringer equation, leading to the rewriting of expressions.
Two different expressions for the Hamiltonian were obtained from calculating the product of ladder operators.
The Hamiltonian allows for rewriting expressions, leading to simplification by distributing the H Omega operator.
The action of the Hamiltonian on S results in E * S, aiding in simplifying the equations.
The Schringer equation for a wave function is akin to a+ S, with a+ being a combination of operators.
Applying the ladder operators repeatedly generates an infinite number of solutions with varying energies.
Solutions with very low energy lead to wave functions curving away from the axis, posing a problem.
The necessity of a normalizable wave function leads to the concept of a lowest energy state, S sub Z.
Solving the ordinary differential equation for the ground state results in the wave function S sub 0.
The raising operator A+ can be applied to construct an infinite number of states with higher energies.

38:33

Solving Equations with Change of Variables

Change of variables is used to simplify equations by eliminating constants.
The desired change of variables involves using x as the square root of H bar over M Omega times a new coordinate C.
Substituting new coordinates requires adjusting the function s from being a function of X to being a function of C.
The second partial derivative with respect to X of s, now a function of C, necessitates the chain rule for differentiation.
The resulting differential equation after substitutions involves the second partial derivative of s with respect to C.
Seeking an asymptotic solution for large C values leads to an approximate solution for the wave function.
Normalizability of the wave function requires B to equal zero, leading to the approximate solution for large C values.
To remove asymptotic behavior, a new function H is introduced as a function of C, leading to a new differential equation.
The new differential equation is simplified by factoring out the exponential dependence, resulting in a transformed ordinary differential equation.
The transformed equation is then solved using power series, with the solution being a power series with unknown coefficients.

55:56

Infinite Power Series Solutions for Differential Equations

The infinite sum of c^2 + A3C^3 continues indefinitely, represented as a sum from J equals 0 to Infinity of a sub J * c to the J.
To substitute this sum into a differential equation, first and second derivatives of the power series need to be calculated.
Derivatives of each term in the power series are found by bringing down the exponent and adjusting coefficients accordingly.
The second derivative is calculated by summing the derivatives of each term, bringing down the exponent by 2.
Substituting the power series into the differential equation involves replacing H, its derivative, and second derivative with their respective power series forms.
To simplify the equation, the power of c in each term of the sum needs to be made consistent by adjusting the indexes.
The coefficients of the power series are determined by setting the terms within the square brackets to zero, leading to a recurrence relation.
The recurrence relation provides a formula for determining coefficients in terms of each other, allowing for the calculation of subsequent coefficients.
The power series solution to the differential equation requires knowledge of two free parameters, a sub 0 and a sub 1, akin to initial conditions in other differential equations.
The power series must terminate to avoid convergence issues, with the highest power determined by a condition on the numerator in the recurrence relation, leading to quantization and restrictions on the power series structure for normalization.

01:13:57

Quantum Harmonic Oscillator Energy Levels and Solutions

The separation constant E is related to energy, given by H bar Omega times n + 0.5.
The quantization condition arises from the requirement for power series termination.
Specific energies are allowed for normalizable solutions in the context of the Schrödinger equation.
Energy levels affect the curvature of wave functions towards or away from the axis.
Incorrect energy levels lead to non-normalizable solutions with specific behaviors.
The power series solutions for the quantum harmonic oscillator involve terms like a0, A1, A2, etc.
The ground state solution, H0, is a simple polynomial, while the wave function S0 is useful.
The first excited state, H1, and wave function S1 are determined by terminating the series at A1.
The second excited state, H2, and wave function S2 are calculated by terminating the series at A2.
The Hermit polynomials have specific properties, including orthogonality and normalization conditions.

01:31:26

"Wave Function Visualization and Propagation Analysis"

The function e to the i kx represents rotation in the complex plane.
Visualizing the function in three dimensions shows a spiral around the zero point.
The function as a function of both position and time results in a spiral that shifts in the x-direction as time increases.
The argument x - h k/2 m t determines the direction of wave propagation.
Setting the constant value of the argument to zero indicates the wave moves in the plus x direction as time increases.
The wave propagates in the opposite direction when the relative sign of x and t is different.
The free particle's lack of boundaries leads to no quantization of energy levels.
Normalization of states for the free particle is challenging due to the lack of boundaries.
Superposition of traveling wave solutions can create wave packets that are localized in space.
Constructing wave packets involves summing up non-normalizable states to form a physically realizable state.

01:50:34

Fourier Analysis: Key Concepts and Applications

Integral from minus infinity to Infinity of DK is followed by F of K, which can be pulled out of the integral over X.
The integral from minus infinity to Infinity DX of e to the minus i k Prime x e to the i kx requires an orthogonality condition for the integral to collapse, resulting in a direct Delta function.
The direct Delta function is defined as a distribution that is nonzero only at a specific value, simplifying the integral to evaluate to F of K Prime.
F of K Prime can be expressed as the integral from minus infinity to Infinity of e to the minus i k Prime x times s of x integral DX, a key concept in Fourier analysis.
Fourier analysis involves expressing a function of X as an integral of a function of K multiplied by e to the i kx, with a symmetry between the Fourier transform equations.
Fourier transforms are crucial in various fields like image processing, where spatial frequency analysis can differentiate features in images effectively.
In quantum mechanics, expressing wave functions as functions of position and time involves determining F of K through integrals, leading to the overall wave function.
Wave velocity in quantum mechanics is determined by the argument of the traveling wave solution, showcasing differences in velocity calculations compared to classical mechanics.
Understanding wave packets involves analyzing the sum of traveling waves with different K values, leading to the concept of slowly varying envelopes and wave packet velocities.
The wave packet velocity is distinct from the velocity of individual features on the wave, with calculations showing differences between classical and quantum mechanics in velocity determinations.

02:11:04

Wave packet velocity and energy spread implications

The velocity for a wave packet is approximately twice the average energy divided by mass in the square root.
Wave packets do not have a single energy, leading to a spread in energies and velocities.
Different parts of a wave packet propagate at varying speeds due to energy spread.
Features on waves can propagate at different speeds than the overall wave packet.
Group velocity and phase velocity differ, impacting wave behavior.
Understanding group and phase velocities is crucial in fields like radio astronomy.
Fourier analysis allows superposition of traveling wave solutions to create localized wave packets.
Initial conditions expressed as a superposition of stationary states aid in wave function evolution.
The Dirac Delta function is a limit of a distribution, becoming infinitely narrow and tall.
Integrating with Delta functions allows for calculating expected values of functions near specific points.

02:28:51

Delta Functions: Extracting Values with Precision

The expected value of f of 0 is independent of the distribution, yielding f of zero.
The infinitely narrow distribution effectively extracts the value of F of x at that point.
The first useful formula with Delta functions is integrating Delta of x times any function f of x, resulting in F of 0.
Delta functions make integrals disappear, applicable not just for Delta function of x but also for Delta functions of x minus a.
Translating the Delta function by a distance a effectively pulls out the value of F at the point a.
Evaluating the Delta function of a function involves integrals multiplied by another function, focusing on where the function crosses zero.
The slope of G of x as it crosses the x-axis determines the effective width of the Delta function.
Expressing Delta of G of x as a sum over the zeros of G of x involves evaluating the effective derivative at those points.
Transforming the Delta function into a sum of shifted Delta functions allows for evaluating the function at specific points.
Multiplying a function by the derivative of the Delta function pulls out the derivative of the function at the argument of the Delta function.

02:46:40

Delta Function and Fourier Transform Relationship

The function capital F of K represents the k-space representation of f of x, with the integral of Delta of X being 1 over 2 pi times e to the -iK.
The Fourier transform of the Delta function is a function of K and x, with the integral involving oscillations between positive and negative values, ultimately equating to zero if X is not equal to x0t.
When X equals x0t, the function becomes constant, leading to an integral of 1 from -infinity to infinity, resulting in infinity, showcasing the relationship between Delta functions and Fourier transforms.

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