So, wave functions are represented by vectors and operators by matrices, all in the space of orthonormal functions. * Example: The Harmonic Oscillator Hamiltonian Matrix. * * Example: The harmonic oscillator raising operator. * * Example: The harmonic oscillator lowering operator. * Now compute the matrix for the Hermitian Conjugate of an operator. The Finite Square Well (Optional) The Quantum Oscillator 212 Expectation Values 217 Observables and Operators 221 209 Quantum Uncertainty and the Eigenvalue Property (Optional) 222 Summary Atomic Hydrogen and Hydrogen-like Ions 277 The Ground State of Hydrogen-like Atoms 282 Excited States of Hydrogen-like Atoms 284 186 Charge-Coupled Devices ...

Remember that ˆa† is just a diﬀerential operator acting on wave functions. Check that you can reproduce the wave functions for the ﬁrst and second excited states of the harmonic oscillator. 12.5 Summary As usual, we summarize the main concepts introduced in this lecture. • Raising and lowering operators; factorization of the Hamitonian.

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