Unitary transformation (quantum mechanics)

In quantum mechanics, the Schrödinger equation describes how a system changes with time. It does this by relating changes in the state of the system to the energy in the system (given by an operator called the Hamiltonian). Therefore, once the Hamiltonian is known, the time dynamics are in principle known. All that remains is to plug the Hamiltonian into the Schrödinger equation and solve for the system state as a function of time.^[1]^[2]

Often, however, the Schrödinger equation is difficult to solve (even with a computer). Therefore, physicists have developed mathematical techniques to simplify these problems and clarify what is happening physically. One such technique is to apply a unitary transformation to the Hamiltonian. Doing so can result in a simplified version of the Schrödinger equation which nonetheless has the same solution as the original.

^ Sakurai, J. J.; Napolitano, Jim J. (2014). Modern Quantum Mechanics (Indian Subcontinent Version ed.). Pearson. pp. 67–72. ISBN 978-93-325-1900-8.
^ Griffiths, David J. (2005). Introduction to Quantum Mechanics (Second ed.). Pearson. pp. 24–29. ISBN 978-0-13-191175-8.

[1] Sakurai, J. J.; Napolitano, Jim J. (2014). Modern Quantum Mechanics (Indian Subcontinent Version ed.). Pearson. pp. 67–72. ISBN 978-93-325-1900-8.

[2] Griffiths, David J. (2005). Introduction to Quantum Mechanics (Second ed.). Pearson. pp. 24–29. ISBN 978-0-13-191175-8.

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