Mathematicians vs. Physics Classes be like...
TLDRIn this humorous and educational video, the transcript showcases a fictional classroom discussion on quantum mechanics and linear algebra. The professor, referred to as 'Chatting,' is criticized for his unrigorous approach to teaching, leading to a series of student objections and humorous exchanges. The video also promotes the use of external tools like Brilliant.org for a more rigorous and active learning experience. The script ends with a call to action for viewers to support the channel and try out Brilliant.org for free.
Takeaways
- 🔬 The video discusses extracting energies from the Schrödinger equation by taking the second derivative of the wave function with respect to position.
- 📚 A student inquires about differentiating the absolute value of x with respect to x, which is not differentiable everywhere, particularly at zero.
- 😅 There is a humorous moment when a student is mistakenly addressed as 'a virgin' instead of 'Mr. Dodson'.
- 🤔 The video touches on the difficulty of finding analytic solutions to non-linear differential equations in physics and the common use of approximations.
- 📉 The concept of using Taylor series to approximate functions is introduced, with the harmonic approximator taking sine of theta equal to theta as an example.
- 🎓 A student expresses frustration with the lack of rigor in the course, highlighting the difference between the course content and the expectations set by Wikipedia.
- 🧠 The video explains the use of matrices in physics to represent systems and the process of inverting matrices to understand system responses.
- 📊 The importance of square matrices with a non-zero determinant for having a real multiplicative inverse is mentioned, with a playful jab at a Mr. Andrews.
- 🤓 A student challenges the validity of an interchange of limits in proving a probability current equation, pointing out the potential issues with justifying the integral.
- 😤 A heated exchange occurs between a professor and a student over the rigor of the class, leading to the student being asked to see the professor after class.
- 👨🏫 The video also covers the role of a function in driving or damping a system in quantum mechanics, with an expansion in a Taylor series as an example.
Q & A
What is the Schrodinger equation and why is it significant in quantum mechanics?
-The Schrodinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is significant because it allows us to calculate the probability distribution of a particle's position and momentum, which are key to understanding quantum phenomena.
How is the second derivative of a wave function related to extracting energies in the Schrodinger equation?
-The second derivative of a wave function with respect to position is related to the kinetic energy of the system. By taking the second derivative, we can find the potential and total energy of the system, which are essential for solving the Schrodinger equation and determining the energy eigenvalues.
Why is the absolute value of x not differentiable at zero, and what does this imply for the wave function?
-The absolute value function, |x|, is not differentiable at x = 0 because it has a sharp corner there, making the slope undefined. This implies that wave functions involving |x| may not be differentiable at certain points, which is a crucial consideration when applying mathematical operations to them.
What is the role of the probability amplitude in quantum mechanics?
-The probability amplitude in quantum mechanics is a complex number that, when squared, gives the probability density of finding a particle in a particular state. It is fundamental in quantum mechanics because it allows us to calculate probabilities of different outcomes, which is not possible in classical physics.
How can non-linear differential equations in physics be approximated if analytic solutions are not available?
-When analytic solutions to non-linear differential equations are not available, physicists often resort to approximations. This can involve using a truncated Taylor series, where higher order terms are dropped, to simplify the equation and make it solvable.
What is the harmonic approximation and how is it used in physics?
-The harmonic approximation is a method used to approximate the behavior of a system by assuming that the restoring force is proportional to the displacement, which is a characteristic of harmonic motion. It simplifies the analysis by treating sine of theta as equal to theta, which is valid for small angles.
Why is the concept of matrix inversion important in physics?
-Matrix inversion is important in physics because it allows us to determine the cause of an observed effect or to predict the response of a system to a given input. It is particularly useful in linear algebra and quantum mechanics, where systems can be represented by matrices.
What are the conditions for a matrix to have a real multiplicative inverse?
-A matrix has a real multiplicative inverse if it is a square matrix (same number of rows and columns) and has a non-zero determinant. These conditions ensure that the inverse exists and is unique.
What is the probability current in quantum mechanics and how is it related to the probability amplitude?
-The probability current in quantum mechanics is a measure of the flow of probability associated with a quantum state. It is related to the probability amplitude through the continuity equation, which ensures that the total probability is conserved over time.
What is the significance of the interchange of limits in the context of the probability current equation?
-The interchange of limits is significant because it allows us to derive the probability current equation from the Schrodinger equation. It is a mathematical operation that requires justification, as it may not always be valid, especially when dealing with improper integrals or non-uniform convergence.
How does the Taylor series expansion of a function relate to its differentiability?
-A function that is infinitely often differentiable is expected to have a convergent Taylor series expansion about any point in its domain. However, differentiability does not guarantee that the function can be represented by a Taylor series everywhere, especially at points of non-analyticity.
What is the role of external tools like Wikipedia, Math Tech Exchange, or Brilliant.org in learning and practicing mathematics?
-External tools like Wikipedia, Math Tech Exchange, and Brilliant.org provide valuable resources for learning and practicing mathematics. They offer a wide range of topics, active learning through exercises, community engagement, and the ability to explore and solve problems at one's own pace.
How does Brilliant.org support active learning in mathematics and physics?
-Brilliant.org supports active learning by offering a variety of topics in mathematics and physics, along with exercises and problems that users can solve. It encourages users to engage with the material, learn by doing, and even participate in a community of learners to enhance their understanding.
Outlines
😖 Confusion in Quantum Mechanics and Linear Algebra
The first paragraph deals with the complexities of solving the Schrödinger equation for stationary states and extracting energies. It highlights the challenge of differentiating absolute values and the common practice of approximating functions in physics by neglecting higher order terms in Taylor series. The discussion also touches on the limitations of matrix inversion and the role of probability in quantum mechanics. The conversation is peppered with classroom interruptions and disagreements about mathematical rigor, suggesting a lack of clarity and consensus on the subject matter.
😡 Frustration with Unrigorous Teaching and Seeking Alternatives
The second paragraph expresses the narrator's frustration with Professor Chatting's unrigorous approach to teaching mathematics and physics. The student resolves to maintain their mathematical skills through self-study and the use of external resources like Brilliant.org, Wikipedia, and Math Tech Exchange. The paragraph transitions into a sponsored promotion for Brilliant.org, an online learning platform offering a wide range of topics and active learning through exercises and community engagement. The promotion includes an offer for free access and a discount for the first 200 users, positioning it as a valuable resource for those seeking a more rigorous learning experience.
Mindmap
Keywords
💡Schrodinger Equation
💡Wave Function
💡Differentiability
💡Probability Amplitude
💡Non-linear Differential Equations
💡Taylor Series
💡Harmonic Approximator
💡Matrix Inversion
💡Probability Current
💡Interchange of Limits
💡Analytic Function
Highlights
Dealing with stationary states in the Schrödinger equation and extracting energies involves taking the second derivative of the wave function with respect to position.
Differentiating the absolute value of x with respect to x is not possible everywhere, specifically not at zero.
The square of the probability amplitude is discussed, highlighting a humorous interaction in the classroom.
Analytic solutions to non-linear differential equations in physics are rare, often requiring approximations by dropping higher order terms in the Taylor series.
Harmonic approximator simplifies sine of theta to theta, illustrating a common approach to handling non-linear problems.
Importance of rigor in mathematics is questioned in the context of physics applications.
Systems in physics can often be expressed in terms of matrices, allowing for analysis of system responses and inverse problems.
Invertibility of matrices in physics is contrasted with the mathematical requirement for a non-zero determinant in square matrices.
Probability in quantum mechanics is discussed, with an interval specification and the introduction of probability current.
Interchange of limits and the existence of integrals are debated in the context of proving a quantum mechanics equation.
The role of functions in driving or damping systems in physics is explained, with an example of expanding an exponential function in a Taylor series.
The concept of a function being infinitely differentiable but not analytical is introduced, leading to a discussion on the rigor of mathematical approaches.
A student expresses frustration with the perceived lack of rigor in the class and the contrast between mathematical and physical approaches.
The use of external tools for mathematical exploration, such as Wikipedia, Math Tech Exchange, and Brilliant.org, is recommended.
Brilliant.org is highlighted as an online learning platform with a wide variety of topics and active learning through exercises.
A sponsored message promotes Brilliant.org, offering free access and a discount for the first 200 users through a provided link.
The video concludes with a humorous interaction between the 'physics boy' and the 'maths boy', emphasizing the fun aspect of filming educational content.
Transcripts
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