Schrodinger equation | Derivation and how to use it

TL;DR

Derivation and use of the Schrodinger equation in quantum mechanics, explaining conservation of energy and the evolution of quantum states over time.

Transcript

this video is the Schrodinger equation part two so if you haven't seen it you might want to watch part one here first on the other hand if you're just here for me to explain the derivation of the equation and how to use it stick around okay so if it was your task to build a new theory of physics you'd need to decide what things are so fundamental t... Read More

Key Insights

• 🦾 The Schrodinger equation is a fundamental tool in quantum mechanics, incorporating conservation of energy and determining the evolution of quantum states over time.
• 👻 It allows for the representation of a particle's state as a superposition of different energy eigenstates, with complex coefficients determining probabilities.
• ❓ The Schrodinger equation explains the varying probability distribution of a particle in different bases, highlighting the significance of rotations and combinations of energy eigenstates.

Questions & Answers

Q: How does the Schrodinger equation incorporate the conservation of energy in quantum mechanics?

The Schrodinger equation ensures conservation of energy by requiring that the energy of a particle remains the same during time evolution, both in the case of an energy eigenstate and in the average energy of the particle. It also ensures that probabilities of energy measurement outcomes remain the same before and after time evolution.

Q: Why does the Schrodinger equation involve complex numbers and rotations?

Complex numbers are used in the Schrodinger equation to represent the coefficients of the energy eigenstates, which determine the probabilities of measuring different energies. Rotations of these complex numbers indicate changes in the angles of the coefficients, which can lead to different probabilities and changes in the overall wave function.

Q: How does the Schrodinger equation explain the varying probability distribution in different bases?

In different bases, such as the position basis, the Schrodinger equation allows for changes in the overall wave function due to rotations and combinations of energy eigenstates. This can result in significant variations in the probability distribution over time, even though the probabilities of energy measurement remain constant.

Q: How do we determine the energy eigenbasis in the Schrodinger equation?

Finding the energy eigenbasis involves solving the equation that defines an energy eigenstate. However, this equation is often mathematically challenging to solve, especially for systems with more than two particles. Physical chemists rely on computational methods to obtain approximate solutions.

Summary & Key Takeaways

• The Schrodinger equation is a fundamental equation in quantum mechanics that incorporates the conservation of energy and describes the evolution of quantum states over time.

• The equation allows us to write the state of a particle as a superposition of different energy eigenstates, and it ensures that energy is conserved during time evolution.

• The Schrodinger equation also introduces the concept of probability in quantum mechanics, where the coefficients of the energy eigenstates determine the probabilities of different measurement outcomes.