Guided reading
The harmonic-oscillator Hamiltonian can be written in terms of two operators. Their action changes the quantum number by one unit, and repeated application of the raising operator constructs the stationary eigenfunctions from the ground state.
Rewriting the Hamiltonian
With \(\hat\xi=\alpha\hat x\) and \(\alpha=\sqrt{m\omega/\hbar}\), define
\[\hat a_+=\frac{1}{\sqrt2}\left(-\frac{d}{d\xi}+\hat\xi\right),\qquad \hat a_-=\frac{1}{\sqrt2}\left(\frac{d}{d\xi}+\hat\xi\right).\]
The Hamiltonian then takes the compact form
\[\hat H=\hbar\omega\left(\hat a_+\hat a_-+\frac12\right).\]
Number operator
For an energy eigenfunction \(\hat H\psi_n(x)=E_n\psi_n(x)\), define
\[\hat N=\hat a_+\hat a_-,\qquad \hat N\psi_n(x)=n\psi_n(x).\]
Thus the oscillator energy \(E_n=\hbar\omega(n+1/2)\) is associated with the eigenvalue \(n\) of \(\hat N\).
Properties used in the construction
\[[\hat a_-,\hat a_+]=1,\qquad [\hat H,\hat a_\pm]=\pm\hbar\omega\hat a_\pm,\qquad [\hat N,\hat a_\pm]=\pm\hat a_\pm.\]
Also, the operators are non-Hermitian conjugates of one another: \((\hat a_\pm)^\dagger=\hat a_\mp\).
Effect on the eigenfunctions
The functions \(\hat a_\pm\psi_n\) are themselves eigenfunctions of both \(\hat H\) and \(\hat N\):
\[\hat H(\hat a_\pm\psi_n)=(E_n\pm\hbar\omega)(\hat a_\pm\psi_n),\qquad \hat N(\hat a_\pm\psi_n)=(n\pm1)(\hat a_\pm\psi_n).\]
Thus applying \(\hat a_+\) raises the energy by one quantum \(\hbar\omega\) and raises the number-operator eigenvalue by one. Applying \(\hat a_-\) lowers the energy by \(\hbar\omega\) and lowers that eigenvalue by one. For this reason, \(\hat a_+\) is the raising operator and \(\hat a_-\) is the lowering operator.
Coefficients of the ladder action
Since the operators change the quantum state by one unit, write \(\hat a_+\psi_n=c_n\psi_{n+1}\) and \(\hat a_-\psi_n=d_n\psi_{n-1}\). Normalization gives \(d_n=\sqrt n\) and \(c_n=\sqrt{n+1}\):
\[\hat a_-\psi_n=\sqrt n\,\psi_{n-1},\qquad \hat a_+\psi_n=\sqrt{n+1}\,\psi_{n+1}.\]
Eigenstates from the ground state
Successive application of the raising operator gives \((\hat a_+)^n\psi_0=\sqrt{n!}\,\psi_n\), hence
\[\psi_n=\frac{(\hat a_+)^n}{\sqrt{n!}}\psi_0.\]
For the ground state, \(\hat a_-\psi_0=0\) gives a first-order differential equation. Its normalized solution is
\[\psi_0(x)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\exp\!\left(-\frac{m\omega}{2\hbar}x^2\right).\]
Connection with the Hermite solution
The operator construction gives the same ground-state wave function and the same stationary eigenfunctions as the solution based on the Hermite differential equation. It is an alternative mathematical route to the oscillator states.
Exercise-ready boundary
This page is designed to support short guided exercises on: the operators \(\hat a_\pm\), the number operator, their commutators, and their action on \(\psi_n\).
- Use from this page: the definitions, physical setup, highlighted equations and conceptual links needed to start a first calculation or explanation.
- Keep in the book: complete derivations, extended historical discussion, worked solutions and the full textbook narrative remain in the original chapter and linked book resources.
- Good exercise balance: ask the student to identify assumptions, apply one relation, and interpret the result physically, without requiring material not introduced on this page.
Practice anchors
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.
- Focus: the raising and lowering construction of oscillator eigenfunctions.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[\hat H=\hbar\omega\left(\hat a_+\hat a_-+\frac12\right)\]
- Equation: \[\hat a_-\psi_n=\sqrt n\,\psi_{n-1},\qquad \hat a_+\psi_n=\sqrt{n+1}\,\psi_{n+1}\]
- Boundary: use this page for setup and first-step reasoning; cite the book for longer derivations, complete experimental history or solved-problem detail.
- Typical task: derive, interpret, or apply the relation above to a simple case without introducing topics outside this page.
Original book and previews: Source note: Original auxiliary summary for this book-app, based on Chapter 4 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc.; consult the original book for the complete presentation, derivations and exercises.