Imagine a quanton in a box whose wave function at time t=0 is a superposition of
ID: 1913357 • Letter: I
Question
Imagine a quanton in a box whose wave function at time t=0 is a superposition of the box's n-th energy eigenfunction?En(x) and its m-th eigenfunction?Em(x) : ?(x)= A?Em(x) + B?En(x)
where m and n are integers such that n>m , and A and B are real constants.
a) the time-evolution rule implies then what is the quanton's wavefunction at time t? write down.
b) show that the absolute square of the wavefunction from (a) is NOT time-independent , but instead has the form f(x)+g(x)cos wt , where the angular frequency of oscillation w=(En - Em) /h
Explanation / Answer
Let the wavefunction in nth energy eigenstate denoted by (phi_n (x))
Given in the question is wavefunction of particle
(psi (x,t=0)=Aphi_m (x)+Bphi_n (x))
(a) the time evolution operator implies that nth energy eigenstate evolves with time and acquires a phase factor of (phase=e^{-iE^nt/hbar})
Thus, at time t, the wavefunction will be:
(psi(x,t)=Aphi_m(x)e^{-iE_mt/hbar}+Bphi_n(x)e^{-iE_nt/hbar})
(b)we can write this to be
(psi(x,t)=e^{-iE_mt/hbar}(Aphi_m(x)+Bphi_n(x)e^{-i(E_n-E_m)t/hbar})=e^{-iE_mt/hbar}(Aphi_m(x)+Bphi_n(x)e^{-iomega t}))
Thus, absolute square of the wavefunction is:
(|psi(x,t)|^2=psi^*psi=(Aphi_m(x)^*+Bphi_n(x)^*e^{iomega t})(Aphi_m(x)+Bphi_n(x)e^{-iomega t})=A^2|phi_m|^2+B^2|phi_n|^2+AB(phi_m^*phi_ne^{-iomega t}+phi_n^*phi_me^{iomega t}))
Since (phi_n) and (phi_m) are real for a particle in a box, we get
(|psi(x,t)|^2=A^2phi_m^2+B^2phi_n^2+ABphi_mphi_n(e^{-iomega t}+e^{iomega t}))
Now, ((e^{-iomega t}+e^{iomega t})=2cos(omega t))
Thus,
(|psi(x,t)|^2=A^2phi_m^2+B^2phi_n^2+2ABphi_mphi_ncos(omega t)=f(x)+g(x)cos(omega t))