The quantum virial theorem
📚 Evaluating expectation values in quantum mechanics typically requires lengthy maths. For example, if we work in the position representation in terms of wave functions, evaluating expectation values typically requires evaluating lengthy integrals. In this video we derive the virial theorem, which provides a very simple relation between the expectation value of the kinetic and potential energies of any quantum system. Such a relation allows us to bypass the lengthy evaluation of expectation values and learn about our system in a simple and transparent manner. We also discuss the special case of potentials described by homogeneous functions, in which the virial theorem takes an even simpler form. Examples we discuss include the quantum harmonic oscillator, the hydrogen atom, and also molecular systems and materials made of many atoms.
0:00 Intro
0:37 Hypervirial theorem
2:57 Virial theorem for 1 particle in 1D
9:13 Virial theorem for potentials described by homogeneous functions
15:25 Virial theorem for multiple particles in 3D
22:29 Wrap-up
⏮️ BACKGROUND
Eigenvalues and eigenstates: • Eigenvalues and eigenstates in quantum mec...
Hermitian operators: • Hermitian operators in quantum mechanics
Expectation values: • Expectation values in quantum mechanics
Commutator algebra: • Commutator algebra in quantum mechanics
Functions of operators: • Functions of operators in quantum mechanics
Quantum harmonic oscillator playlist: • Quantum harmonic oscillator
Hydrogen atom playlist: • Hydrogen atom
⏭️ WHAT NEXT?
Hydrogen atom ground state: [COMING SOON]
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Director and writer: BM
Producer and designer: MC