and get an eigenvalue ħk for a momentum. Nonetheless, such y(x) does not satisfy
the proper BCs; i.e., y(L) ¼ y(ÀL ) ¼ 0. This is because e
ikx never vanishes with any
real numbers of k or x (any complex numbers of k or x, more generally). Thus, we
cannot obtain a proper solution that has an eigenstate with a fixed momentum in a
confined physical system.
References
1. Messiah A (1999) Quantum mechanics. Dover, New York
2. Stakgold I (1998) Green’s functions and boundary value problems, 2nd edn. Wiley, New York
3. Lebedev NN (1972) Special functions and their applications. Dover, New York
4. Shimizu A (2003) Upgrade foundation of quantum theory (in Japanese). Saiensu-sha, Tokyo
56
2 Quantum-Mechanical Harmonic Oscillator
the proper BCs; i.e., y(L) ¼ y(ÀL ) ¼ 0. This is because e
ikx never vanishes with any
real numbers of k or x (any complex numbers of k or x, more generally). Thus, we
cannot obtain a proper solution that has an eigenstate with a fixed momentum in a
confined physical system.
References
1. Messiah A (1999) Quantum mechanics. Dover, New York
2. Stakgold I (1998) Green’s functions and boundary value problems, 2nd edn. Wiley, New York
3. Lebedev NN (1972) Special functions and their applications. Dover, New York
4. Shimizu A (2003) Upgrade foundation of quantum theory (in Japanese). Saiensu-sha, Tokyo
56
2 Quantum-Mechanical Harmonic Oscillator