# Lecture 7: More on Energy Eigenstates

TLDRThis lecture delves into the foundational concepts of quantum mechanics, focusing on the properties and behaviors of energy eigenfunctions. The professor elucidates the relationship between observables and operators, introduces the notion of eigenfunctions, and discusses the implications of degeneracy. The lecture also explores the time evolution of quantum states and the significance of the Schrödinger equation. Through examples, such as the infinite square well, the professor illustrates how energy eigenfunctions can be used to describe systems with bound states and the emergence of discrete energy levels.

###### Takeaways

- 📚 The lecture discusses the concept of the inner product for functions, analogous to the dot product for vectors, which is a method to extract a scalar from two functions.
- 🌟 The script explains the basic postulate of quantum mechanics that associates observables with operators acting on wave functions, and the importance of eigenfunctions in describing states with definite values of observables.
- 🚀 The professor emphasizes the orthogonality and normalization of eigenfunctions, particularly for position, momentum, and energy, which are essential in quantum mechanics.
- 🔍 It is clarified that eigenfunctions corresponding to different eigenvalues are orthogonal, and the concept of degenerate eigenfunctions having the same eigenvalue is introduced.
- 📈 The script explores the possibility of expanding any function in a basis of eigenfunctions of an observable, which is a statement derived from the spectral theorem.
- 🌱 The difference between discrete and continuous energy levels is discussed, with the former being associated with bound states and the latter with states that can extend infinitely.
- ⏳ The time evolution of quantum states is described, with energy eigenfunctions evolving with a simple phase factor due to their special properties from the Schrödinger equation.
- 💡 The professor uses the example of an infinite square well to illustrate the discrete energy levels and the corresponding wave functions, highlighting the differences from classical intuition.
- 🔑 The script touches on the Heisenberg uncertainty principle by discussing the non-commutativity of position and momentum operators.
- 📊 The lecture includes an interactive session with clicker questions to test students' understanding of eigenstates, superpositions, and the physical interpretation of quantum states.
- 🤔 The importance of qualitative understanding in quantum mechanics is stressed, encouraging students to develop an intuition for the behavior of wave functions in various potentials.

###### Q & A

### What is the notation used to represent the inner product for functions and how is it related to the dot product of vectors?

-The notation used to represent the inner product for functions is the bracket symbol, such as `<f, g>`, which signifies the integral of the complex conjugate of function f multiplied by function g over all positions. It is analogous to the dot product of vectors, where you get a scalar value from two vectors by taking their components and multiplying them together.

### What is the physical interpretation of eigenfunctions in quantum mechanics?

-Eigenfunctions in quantum mechanics are special functions that, when acted upon by an operator corresponding to an observable, return the same function times a constant. Physically, these eigenfunctions represent the wave functions describing configurations with a definite value of the corresponding observable.

### How do the eigenfunctions of position and momentum relate to the states of a quantum system?

-The eigenfunctions of position are delta functions, representing states with a definite position. The eigenfunctions of momentum are exponentials, representing states with a definite value of momentum. If a system is described by an eigenfunction of the position operator, it means the system is in a configuration with a definite position. Similarly, if described by an eigenfunction of the momentum operator, it indicates a state with a definite momentum.

### What is the significance of orthogonality in the context of eigenfunctions?

-Orthogonality in the context of eigenfunctions means that eigenfunctions corresponding to different eigenvalues are different functions and are orthogonal in the sense of the inner product. This property ensures that the eigenfunctions can form a complete set, allowing any function to be expanded as a superposition of these eigenfunctions.

### What does it mean for two eigenfunctions to be degenerate?

-Two eigenfunctions are said to be degenerate if they have the same eigenvalue but are not proportional to each other. This can occur in quantum mechanics, and it implies that there is a degeneracy in the energy levels, allowing for additional quantum states beyond the non-degenerate case.

### How does the spectral theorem relate to the expansion of any function in terms of eigenfunctions?

-The spectral theorem states that it is always possible to find a set of eigenfunctions which are orthogonal, even in the presence of degeneracies. This allows any function to be expanded in a basis of these eigenfunctions, with each eigenfunction corresponding to a definite value of an observable.

### Why are energy eigenfunctions particularly useful in quantum mechanics?

-Energy eigenfunctions are particularly useful because they evolve over time in a simple way, changing only by an overall phase factor. This makes them convenient for solving the time evolution problem in quantum mechanics, as any state can be expressed as a superposition of energy eigenfunctions.

### What is the difference between the energy eigenfunctions of a free particle and those of an infinite square well?

-For a free particle, the energy eigenfunctions can be continuous and form a continuous spectrum due to the particle's ability to have any energy value. In contrast, the energy eigenfunctions of an infinite square well are discrete and form a discrete spectrum because the particle is confined within the well and can only have specific quantized energy levels.

### How does the probability distribution of a quantum particle in the ground state of an infinite square well differ from the classical expectation?

-In the ground state of an infinite square well, the quantum particle's probability distribution is not uniform as one might expect classically. Instead, it has a higher probability of being found near the walls of the well, which is a direct result of quantum effects and the wave nature of the particle.

### Why is the energy spectrum of a system with bound states discrete?

-The energy spectrum of a system with bound states is discrete because the states are trapped in a certain region and are not allowed to have arbitrarily large energy values. Each allowed energy level corresponds to a specific state of the system, leading to a discrete set of possible energy eigenvalues.

### What is the relationship between the wave function of a quantum system and its probability distribution?

-The probability distribution of a quantum system is given by the norm squared of the wave function. This means that the probability of finding the system in a particular state is proportional to the square of the magnitude of the wave function at that state.

###### Outlines

##### 📚 Introduction to Quantum Mechanics and Eigenfunctions

The script begins with an introduction to the topic of quantum mechanics, focusing on the concept of eigenfunctions and their significance in representing physical observables. It explains the dot product for functions, which is analogous to the dot product for vectors, and how it is used to calculate a number from two functions. The professor emphasizes the importance of eigenfunctions as they describe states with definite values of an observable, such as position, momentum, and energy. The script also discusses the orthogonality and normalization of these eigenfunctions, which are key properties in quantum mechanics.

##### 🔍 Degeneracy of Eigenfunctions and Spectral Theorem

This paragraph delves into the concept of degenerate eigenfunctions, where two different eigenfunctions can have the same eigenvalue. The professor explains that this is not only possible but also has physical implications. It introduces the idea that while eigenfunctions corresponding to different eigenvalues are orthogonal, degenerate eigenfunctions are not necessarily so. The spectral theorem is mentioned, which states that even with degeneracies, it's possible to find a set of orthogonal eigenfunctions, forming a basis for the space of functions.

##### 🌟 Expansion of Functions in Eigenfunction Basis

The script discusses the ability to expand any function in terms of eigenfunctions of an observable, such as position or momentum. It uses the Fourier theorem to explain the expansion in terms of momentum eigenfunctions and touches on the physical interpretation of these expansions. The paragraph highlights the mathematical and physical significance of expressing any state as a superposition of states with definite values of an observable.

##### ⚛️ Quantum States and Energy Eigenfunctions

The focus shifts to energy eigenfunctions and their unique properties, particularly how they evolve over time. The professor explains that while energy eigenfunctions do not represent states with definite positions or momenta, they do have a simple time evolution described by the Schrödinger equation. The paragraph also addresses the discrete and continuous nature of energy eigenvalues in different physical systems and the implications for the expansion of wave functions.

##### 📉 Time Evolution of Quantum States

This section discusses the time evolution of quantum states, especially those that are not energy eigenfunctions. The professor explains that while energy eigenfunctions evolve with a simple phase factor, a general state's evolution is more complex. The paragraph emphasizes the importance of understanding the time evolution of quantum states for solving problems in quantum mechanics.

##### 📚 Infinite Square Well and Energy Eigenfunctions

The script provides an example of the infinite square well to illustrate the concept of energy eigenfunctions. It describes the potential energy function and how the energy eigenfunctions for this system are sine waves that satisfy the boundary conditions. The paragraph explains the quantization of energy levels and the corresponding wave functions for the ground state and the first excited state.

##### 📈 Probability Distributions and Quantum Effects

This paragraph explores the probability distributions associated with different energy eigenfunctions of the infinite square well. It contrasts the classical expectation of a uniform probability distribution with the quantum mechanical reality, where the probability distribution is not uniform and has nodes. The professor raises questions about the implications of these findings and their relation to the physical properties of materials like diamonds.

##### 🤔 Quantum Mechanics Intuition and Thought Experiments

The script invites students to ponder the differences between classical and quantum mechanical probability distributions, especially at low energies where quantum effects dominate. It challenges students to think about why certain quantum phenomena occur, hinting at the role of superposition and interference in quantum mechanics.

##### 📝 Review and Application of Quantum Concepts

The final paragraph of the script is a review of the concepts discussed and their application to quantum mechanics problems. The professor uses clicker questions to test students' understanding of eigenfunctions, time evolution, and the properties of quantum states. The questions cover topics such as the commutator of position and momentum operators, the inner product, and the behavior of wave functions.

##### 🔬 Energy Eigenfunctions and Qualitative Behavior

The script concludes with a discussion on the qualitative behavior of energy eigenfunctions for an arbitrary potential. It explains how to derive the differential equation for energy eigenfunctions and how to interpret the second derivative in terms of curvature. The professor guides students through the process of estimating the qualitative features of wave functions based on the given potential.

##### 🌐 Curvature Analysis in Energy Eigenfunctions

This section provides a detailed analysis of how the curvature of a wave function, as indicated by the second derivative, can be used to predict the behavior of energy eigenfunctions. It explains that in classically allowed regions, the wave function should be sinusoidal, while in classically forbidden regions, it should be exponential. The professor uses this concept to sketch the expected wave function for a given energy and potential.

##### 🚫 Non-Physical Wave Functions and Boundary Conditions

The script highlights the importance of boundary conditions in determining the physicality of a wave function. It demonstrates that a wave function that diverges to infinity is non-physical and must satisfy conditions such as vanishing at infinity for normalizability. The discussion also touches on the discrete nature of energy levels in systems with classically forbidden regions.

##### 🔚 Conclusion and Future Discussion

The final paragraph of the script wraps up the current discussion and hints at future topics. It leaves students with a teaser about the discrete nature of the hydrogen spectrum and the reasons behind it, setting the stage for the next class.

###### Mindmap

###### Keywords

##### 💡Creative Commons license

##### 💡MIT OpenCourseWare

##### 💡Dot product

##### 💡Complex conjugate

##### 💡Observable

##### 💡Eigenfunctions

##### 💡Position operator

##### 💡Momentum

##### 💡Energy eigenfunctions

##### 💡Orthogonal functions

##### 💡Degeneracy

##### 💡Spectral theorem

##### 💡Fourier theorem

##### 💡Schrodinger equation

##### 💡Infinite square well

##### 💡Quantum superposition

##### 💡Quantum interference

###### Highlights

Introduction to the dot product analogy for functions, integral notation, and its significance in quantum mechanics.

Explanation of the bracket symbol representing the integral and its comparison to the dot product for vectors.

Discussion on the properties of the dot product and its relation to function norms, emphasizing non-negativity.

Quantum mechanics postulate that every observable is associated with an operator acting on wave functions.

Introduction to eigenfunctions and their physical interpretation as wave functions with definite observable values.

Clarification on the orthogonality of eigenfunctions with different eigenvalues and their normalization.

The possibility of degenerate eigenfunctions sharing the same eigenvalue and the implications for physical systems.

Exploration of the spectral theorem and its assertion that eigenfunctions can be orthogonal even with degeneracies.

Expansion of any function in a basis of eigenfunctions with definite values of an observable.

The difference between discrete and continuous energy eigenvalues and their dependence on system properties.

The Schrodinger equation and its implications for the time evolution of energy eigenfunctions.

Illustration of how energy eigenfunctions evolve over time with a simple phase change.

Examples of energy eigenfunctions for a free particle and an infinite square well, highlighting their simplicity and boundary conditions.

Analysis of the probability distributions for the ground state and excited states in an infinite square well.

Discussion on the differences between classical and quantum mechanical probability distributions at low and high energies.

The relationship between quantum effects, superposition, and interference as causes for observed phenomena in quantum mechanics.

Interactive use of clicker questions to engage the audience and assess understanding of eigenstates and wave function properties.

Final remarks on the importance of understanding the qualitative behavior of energy eigenfunctions for various potentials.

###### Transcripts

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