Exploring Quantum Mechanics with Variational Calculus

In summary, a student is taking a Quantum Mechanics course and their professor discussed the concepts of Newtonian Mechanics leading to Lagrangian Mechanics and Hamiltonian Mechanics, eventually leading to Quantum Mechanical wave functions using variational calculus. The student is seeking recommendations for textbooks that cover these topics in detail, with "Advanced Calculus" by Loomis and "Calculus of Variations" by Gelfand and Fomin being suggested. They also mention the importance of understanding linear algebra and suggest looking into the Dirac notation. Additional textbook suggestions are "Classical Mechanics" by Herbert Goldstein and "Mathematical Methods for Physicists" by Arfken.
  • #1
aliaze1
174
1
I am taking a Quantum Mechanics course this semester and the professor started off by showing us how Newtonian Mechanics lead to Lagrangian Mechanics, then Hamiltonian...etc...until we got to the Quantum Mechanical wave functions.

This was all done in a quantitative sense, using variational calculus. Does anyone know of any good mathematics/physics/mathematical physics textbook that goes into this type of detail for mechanics? I was looking into "Advanced Calculus" by Loomis, since Loomis' Calculus book was amazing, I imagine the Advanced book would be good too..
 
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  • #2
you have got a very good teacher. many times such personal viewpoints are impossible to find in any book. and taking lessons from classroom is the only way to get such personal viewpoints. so absorb it as much as you can. discuss with the person too. you will not get a chance later on.
 
  • #4
Of course you'll need some calculus, but I also suggest you get a good understanding of linear algebra. You might also start looking at the Dirac notation (after you study a little linear algebra).
In older QM books you see states expressed in terms of wave functions with integrals, etc. and a lot of calculus. Newer books use more the Dirac notation, which is based on the linear algebra and the fact that you can represent functions in a "Hilbert space"
Books:
"Classical Mechanics" by Herbert Goldstein.
"Mathematical Methods for Physicsists" Arfken.
"Mathematical Methods in the Physical Sciences" is also good. But Arfken is a little more advanced I think. It wouldn't hurt having both.
 

1. What is quantum mechanics and how is it related to variational calculus?

Quantum mechanics is a branch of physics that studies the behavior and interactions of particles at the atomic and subatomic level. Variational calculus is a mathematical method used to find the minimum or maximum value of a function. In quantum mechanics, variational calculus is used to find the most accurate approximation to the true wave function of a quantum system.

2. How does variational calculus help in understanding quantum systems?

Variational calculus allows for the calculation of the ground state wave function of a quantum system, which is the lowest energy state of the system. This information is crucial in understanding the behavior and properties of the system, such as energy levels and probabilities of different outcomes.

3. What are the benefits of using variational calculus in quantum mechanics?

Variational calculus offers a more efficient and accurate approach to understanding quantum systems compared to traditional methods. It also allows for the treatment of complex systems with multiple particles, making it a valuable tool in research and experimentation.

4. Are there any limitations to using variational calculus in quantum mechanics?

While variational calculus is a powerful tool, it can only provide approximations to the true wave function of a quantum system. This means that there may be some degree of error in the calculations, particularly for highly complex systems.

5. How is "Exploring Quantum Mechanics with Variational Calculus" different from other resources on the topic?

This book offers a comprehensive and practical guide to using variational calculus in the context of quantum mechanics. It not only explains the theory and concepts behind the method, but also provides examples and exercises for readers to practice and apply their understanding. It also includes real-world applications and discussions on the current research and developments in the field.

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