Another application of the ladder operator concept is found in the quantum-mechanical treatment of the harmonic oscillator. We can define the lowering and raising operators as a ^ ^ = m ω ω 2 ℏ ℏ ( x ^ ^ + i m ω ω p ^ ^ ) , a ^ ^ † † = m ω ω 2 ℏ ℏ ( x ^ ^ − − i m ω ω p ^ ^ ) .