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Fundamentals of Quantum Physics 2: Superposition. Particle in a box 🌚 Lecture for Sleep & Study

LECTURES FOR SLEEP & STUDY・127 minutes read

The Schrodinger equation in quantum mechanics focuses on solving partial differential equations using separation of variables, leading to stationary states with constant properties and exact energies. Superpositions of these stationary states allow for complex dynamics in wave functions, showing the importance of time-dependent solutions and energy distributions in understanding quantum mechanics.

Insights

Schrodinger equation is the central focus of quantum mechanics, requiring a deep understanding of solving similar equations involving ordinary and partial differential equations.
Separation of variables is a crucial technique for solving partial differential equations, leading to stationary states with constant probability densities and expectation values.
The linearity of the Schrodinger equation allows for constructing complex solutions by superimposing stationary states, showcasing the connection to energy distribution over time.
The potential energy function in the Schrodinger equation significantly impacts the probability distribution of finding a particle, with different potential scenarios influencing the curvature and behavior of wave functions.

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Recent questions

What is the Schrodinger equation?
The Schrodinger equation is the fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time.
What are partial differential equations?
Partial differential equations involve multiple independent variables and describe how functions change with respect to these variables.
How does Gauss's law relate to electric fields?
Gauss's law establishes a relationship between the electric field and the charge enclosed by a surface.
What is the Laplace equation?
The Laplace equation is derived from Gauss's law and relates to the potential in a given system.
What is the wave equation?
The wave equation describes the forces due to curvatures in a string and is a key concept in physics.

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Summary

00:00

Key Equations in Quantum Mechanics Explained

Schrodinger equation is the main focus of quantum mechanics.
To solve the Schrodinger equation, understanding how equations like it are solved is crucial.
Ordinary differential equations describe how specific coordinates change with time.
Partial differential equations involve several independent variables.
Gauss's law relates the electric field to the charge enclosed by a surface.
The Laplace equation is derived from Gauss's law and relates to the potential.
Separation of variables is a key technique for solving partial differential equations.
The wave equation arises from forces due to curvatures in a string.
Separation of variables involves breaking down functions into simpler components.
Solving partial differential equations involves finding constants through boundary and initial conditions.

18:46

"Understanding Schrodinger Equation and Wave Solutions"

Wave equation allows finding traveling wave solutions with appropriate constants.
Heat equation relates to heat diffusion in materials, involving separation of variables.
Time-dependent Schrodinger equation involves Hamiltonian operator and wave function.
Hamiltonian includes kinetic and potential energy operators.
Schrodinger equation is expressed as IHbarDPSI/DT = -Hbar^2/2mD^2PSI/DX^2 + V(x)PSI.
Separation of variables leads to equations for time and space parts.
Time part simplifies to exponential form for time evolution.
Space part involves solving for X with potential V(x) determining wave function.
Hamiltonian operator acting on spatial part equals energy times spatial part.
Separated solutions are stationary states with constant probability density and expectation values.

38:19

"Stationary states and time dynamics in quantum"

Multiplying two operators results in one, with the time part dropping out if the operator has no partial time derivatives.
Expectation values of operators have no time dependence if the wave function separates, leading to constant expectation values.
The energy of a stationary state is expressed as the Hamiltonian operator acting on the spatial part of the wave function, equal to the separation constant times the spatial part.
Calculating the expectation value of the Hamiltonian operator strengthens the connection to the energy of the system.
The variance in energy is zero for stationary states, indicating no uncertainty in energy.
Stationary states have constant probability densities, expectation values, and exact energy, existing indefinitely.
Superpositions of stationary states create non-trivial time dynamics due to the linearity of the Schrodinger equation.
The linearity of the Schrodinger equation allows for constructing solutions by adding solutions together.
The general solution to the Schrodinger equation involves summing solutions to the time independent Schrodinger equation with a time-dependent part.
The linearity of the Schrodinger equation enables the creation of complex solutions by combining individual solutions.

57:43

Quantum Mechanics: Schrodinger Equation and Solutions

The general expression for summing up stationary state solutions to the time-independent Schrodinger equation involves a constant C sub J that determines the contribution of each stationary state.
This solution to the Schrodinger equation is comprehensive, encompassing all possible solutions due to the nature of the Schrodinger equation as a Sturm-Liouville problem.
The Schrodinger equation's classification as a Sturm-Liouville problem enables the superposition of stationary states, allowing for the formation of any desired solution.
By constructing a wave function from distinct solutions to the time-independent Schrodinger equation, one can analyze the energy distribution over time.
The expectation value of the Hamiltonian operator for the wave function involves complex exponential time dependencies and integrals over spatial parts.
The energy expectation value simplifies to C1 squared E1 plus C2 squared E2, showcasing how the energies of the states multiply together.
The integral of the product of distinct solutions like X1 and X2 results in zero, a characteristic of Sturm-Liouville problems.
Quantum mechanics involves solving the Schrodinger equation by expressing initial conditions as a superposition of stationary states and adding time dependence to each state.
The Schrodinger equation's complexity arises from the diverse potential functions that influence the physics of quantum mechanics.
The potential energy function in the Schrodinger equation impacts the probability distribution of finding a particle in regions with varying potential energy levels.

01:17:19

"Potential Energy in Schrodinger Equation Solutions"

The Schrodinger equation relates potential energy to wave function behavior
Multiplying by H bar squared over 2m rearranges terms to relate second derivative of PSI to PSI itself
Potential larger than energy results in positive second derivative, while energy larger than potential yields negative second derivative
Example potential functions include particle in a box, harmonic oscillator, Delta function potential, particle in a soft box, and no potential
Particle in a box potential has infinite energy outside and zero inside, representing a non-physical scenario
Harmonic oscillator potential resembles a parabola, common in classical physics, representing a particle attached to a spring
Delta function potential is infinitely sharp and deep, with different behavior due to lack of infinite potential energy
Particle in a soft box potential has finite energy, representing a box with pores requiring energy to push particles in
No potential scenario has constant potential everywhere, impacting Schrodinger equation solutions
Understanding potential influences on Schrodinger equation solutions involves analyzing curvature and behavior of wave functions, with potential larger than energy leading to upward curvature and vice versa

01:35:52

"Solving Schrodinger Equation in Infinite Square Well"

The Schrodinger equation is solved under specific scenarios, with the first example being the infinite Square well.
The potential in the infinite Square well is infinite outside the region between 0 and a, and zero within that region.
The time-independent Schrodinger equation is utilized, simplifying to a second partial derivative of PSI with respect to X.
Outside the region, the wave function PSI is zero, while inside, the equation simplifies to -H bar squared over 2m times the second partial derivative of PSI with respect to X is equal to e times PSI.
The solution to the equation involves a constant K, defined as root 2m times e over H bar, leading to a general solution of PSI as a function of X being a sine KX plus b cosine KX.
Boundary conditions dictate that the wave function must be continuous, resulting in PSI of 0 being equal to 0 and PSI of a being equal to 0.
The wave function is determined to be a sine of n pi over a times X, with n being an integer, leading to a discrete set of allowed energies.
The energy solutions are found to be N squared pi squared H bar squared over two m a squared, representing a discrete set of allowed energies.
The normalization constant is determined to be the square root of 2 over a, leading to the final normalized wave function as root 2 over a times the sine of n pi over a X.
A diagram is drawn to visualize the discrete set of allowed energies and corresponding wave functions, showing distinct energy levels and wave function shapes.

01:55:38

Quantum wave functions and Fourier analysis

Wave functions gradually increase in energy from 1 to 4 to 9 to 16, following an N squared pattern.
At the midpoint of the interval, wave functions exhibit a pattern of maximum-zero alternation.
Wave functions are either even or odd about the midpoint, alternating between the two.
The symmetry and properties of wave functions connect to Fourier analysis and Fourier Series.
Orthogonality and completeness of solutions to the Schrodinger equation are crucial for mathematical properties.
Stationary states for the infinite Square well potential involve sine functions.
The number of nodes in wave functions indicates points where the function reaches zero.
Orthogonality in quantum mechanics is akin to the dot product of vectors.
Sine functions in the infinite Square well potential satisfy an orthogonality integral.
Fourier's trick involves expressing a function as an infinite sum of constants multiplied by sine functions.

02:14:24

"Expressing Functions as Sums of Sine"

The integral for normalization results in a value of a over 2.
The left-hand side equals CM times a over 2 after integrating all terms.
CM is calculated as 2 over a times the integral from 0 to a of f of x sine M pi x over a DX.
This method allows expressing any function as a sum of sine functions using a specific formula.
The technique enables expressing initial conditions in the Schrodinger equation as a sum of stationary states.
Superpositions of stationary states are solutions to the Schrodinger equation, aiding in constructing wave functions.
Solutions to the time-independent Schrodinger equation form a complete basis set.
The method works for smooth, square-integrable functions, not for functions with infinite values or discontinuities.
The technique involves expressing functions as a sum of sine functions, adjusting the numbers in the sum using Fourier's trick.
The process allows recreating any function by combining sharp spikes at different positions and heights.

02:32:20

"Complex expressions simplified for numerical understanding"

Initial conditions defined as square root of three over a times one minus absolute value expression and integral of G of x times PSI from 0 to a over 2 plus G of x times PSI from a over 2 to a.
PSI stars omitted as evaluation at T equals zero makes them irrelevant.
Integral split from 0 to a over 2 and a over 2 to a for simplicity in Sage.
Plotting G of x as a function of X with a set to one, resulting in a tent function.
Evaluation of C of n by plugging into the integral, providing expressions like 4 times square root of 6 over pi squared.
Approximation of C sub n expressions to understand numerical values better.
Truncating sum at n equals 3 to focus on PSI 1 and PSI 3 for expectation of x squared.
Calculation of expected value of x squared involving complex exponentials and specific functions.
Utilizing Sage to understand time dependence in the expression and simplifying complex parts.
Creation of test function using C sub NS and evaluating it for simplicity, leading to a cosine-like plot.

02:50:14

"Quantum simulation reveals complex wave dynamics"

The simulation involves five stationary states evolving with the same amplitude but different phases, with the ability to introduce a Gaussian into the system to manipulate initial conditions and observe wave function changes over time.
By playing with the simulation, one can see how the superposition of different states leads to complex dynamics in the overall probability density, showcasing the behavior of the Schrodinger equation in various physical scenarios and emphasizing the importance of understanding time evolution and energy differences in quantum mechanics.

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