- #1
vani11a
- 1
- 0
I have some questions concerning quantum optics and quantum mechanics and I'd appreciate if someone could help me with them.(I'm self-studying both so the questions can sound stupid)
1) When treating beam-splitter quantum-mechanically, why we replace classical [tex]E[/tex] amplitudes with annihilation operators [tex]a[/tex] and don't take both [tex]a[/tex] and [tex]a^{+}[/tex](since [tex]E[/tex] is a sum of them)? I can't understand why we treat the field on the 'out' ports with annihilation operators.
2) Is there any physical meaning for 'negative-frequency' part when quantizing electric field: [tex] E^{(-)}=\sum_{k}\epsilon_kE_ka_{k}^{+}e^{i\omega_kt-ikr} [/tex] and why we sometimes use only [tex]E^{(+)}[/tex] or [tex]E^{(-)}[/tex]?
3)Why in down-conversion the Hamiltonian is of the form [tex] \hbar g(a_{1}^{+}a_{2}^{+}a_{0}+h.c.)[/tex]? Should it be derived replacing [tex]P[/tex] with the expression with [tex]\chi^{(2)}[/tex]?
4)A question about QM: What is actually postulated? Statements about correspondence between isolated physical systems and vectors in Hilbert space or between measurables and operators are usually the same in all the literature I've read, but I've also seen postulating the expression for momentum operator or the relation between commutator and Poisson bracket, though in Landau&Lifgarbagez both of those are derived.
Thanks in advance.
1) When treating beam-splitter quantum-mechanically, why we replace classical [tex]E[/tex] amplitudes with annihilation operators [tex]a[/tex] and don't take both [tex]a[/tex] and [tex]a^{+}[/tex](since [tex]E[/tex] is a sum of them)? I can't understand why we treat the field on the 'out' ports with annihilation operators.
2) Is there any physical meaning for 'negative-frequency' part when quantizing electric field: [tex] E^{(-)}=\sum_{k}\epsilon_kE_ka_{k}^{+}e^{i\omega_kt-ikr} [/tex] and why we sometimes use only [tex]E^{(+)}[/tex] or [tex]E^{(-)}[/tex]?
3)Why in down-conversion the Hamiltonian is of the form [tex] \hbar g(a_{1}^{+}a_{2}^{+}a_{0}+h.c.)[/tex]? Should it be derived replacing [tex]P[/tex] with the expression with [tex]\chi^{(2)}[/tex]?
4)A question about QM: What is actually postulated? Statements about correspondence between isolated physical systems and vectors in Hilbert space or between measurables and operators are usually the same in all the literature I've read, but I've also seen postulating the expression for momentum operator or the relation between commutator and Poisson bracket, though in Landau&Lifgarbagez both of those are derived.
Thanks in advance.