Quantum Mechanics
Schrodinger equation
Quantum Mechanics
Propagators : Pg
Quantum Simple Harmonic Oscillator QSHO
Quantum Mechanics
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© The scientific sentence. 2010

 Manyelectron atoms: Noninteracting particles
1. The multiparticle system :
The Hamiltonian of an Nparticle system is:
H(r1, r2, ..., r_{N},t) =
Σ(i) P_{i}^{2}/2m_{i} + V(r1, r2, ..., r_{N},t)
i from 1 to N
The term :
Σ(i) P_{i}^{2}/2m_{i}
represents the total kinetic energy of the system.
The term:
V(r1, r2, ..., r_{N},t)
represents the interaction between the particles
making up the system, and the interaction of these particles
with the outside of the system.
If the particles do not interact with one another, then
V(r1, r2, ..., r_{N},t) will represent only the
interaction of the particles with the outside of the system.
(The outside of the system of N electrons can be the nucleus
of the atom they constitute). Therefore, the common potential
becomes:
V(r1, r2, ..., r_{N},t) = Σ(i) V(ri,t)
i from 1 to N
The total Hamiltonian becomes:
H(r1, r2, ..., r_{N},t) = Σ(i) P_{i}^{2}/2m_{i} +
Σ(i) V(ri,t)
= Σ(i) {P_{i}^{2}/2m_{i} + V(ri,t)} =
Σ(i) H_{i}(ri,t)
i from 1 to N
Where
H_{i}(ri,t) = P_{i}^{2}/2m_{i} + V(ri,t)
Therefore
For noninteracting particles, the Nparticle Hamiltonian of
the system is the sum of N independent singleparticle
Hamiltonians. The energies of the particles making up
the system are independent; and their instantaneous positions are
also not related. Hence, the Nparticle system wave function
ψ(r1, r2, ..., r_{N},t)
is written as the product of Nindependent singleparticle
wavefunctions:
ψ(r1, r2, ..., r_{N},t) =
ψ_{1}(r1,t) ψ_{2}(r2,t) ... ψ_{N}(rN,t) =
Π(i) ψ_{i}(ri,t)
i from 1 to N
Let's recall ψ_{i}(ri,t)^{2}
is the probability of finding the ith particle at the
time t between ri and ri + dri. Since it is independent and
completely unaffected by the others, this
probability must satisfy the following normalization
condition:
∫ψ_{i}(ri,t)^{2} dri = 1
dri from  ∞ to + ∞
The probability of finding the whole Nindependent
particle system at the time t with each ith particle
between its ri and ri + dri is ψ(r1, r2, ..., r_{N},t)^{2};
that must satisfy the following normalization
condition:
∫ψ(r1, r2, ..., r_{N},t))^{2} dr1 dr2 ... dr_{N} = 1
dri from  ∞ to + ∞
The timedependent Schrodinger equation for this whole system of
N noninteracting particles is:
iħ ∂ ψ(r1, r2, ..., r_{N},t)/∂t = H ψ(r1, r2, ..., r_{N},t)
The corresponding multiparticle state of definite energy E is:
ψ(r1, r2, ..., r_{N},t) = ψ_{E}(r1, r2, ..., r_{N}) exp{ iEt/ħ}
ψ(r1, r2, ..., r_{N},t) is the eigenstate of the Hamiltonian H of the system.
ψ_{E}(r1, r2, ..., r_{N}) is the stationary state of the Hamiltonian H of this system.
E is the eigenvalue of the Hamiltonian H of the system.
The stationary wavefunction ψ_{E}(r1, r2, ..., r_{N}) of the system
satisfies the timeindependent Schrodinger equation
Hψ_{E}(r1, r2, ..., r_{N}) = E ψ_{E}(r1, r2, ..., r_{N})
The timedependent Schrodinger equation for this system of
N noninteracting particles is then written as a
set of N independent equations, for
each particle, of the form :
iħ ∂ψ_{i}(ri,t)/∂ = H_{i} ψ_{i}(ri,t)
i from 1 to N
In the case of the potential is timeindependent, we have:
V(ri,t) = V(ri)
ψ(ri,t) = ψ_{E}(ri) exp{ iE_{i}t/ħ}
ψ(ri,t) is the eigenstate of the Hamiltonian H_{i} of one particle.
ψ_{E}(ri) is the stationary state of the Hamiltonian H_{i} of this particle.
E_{i} is the eigenvalue of the Hamiltonian H_{i} of this particle.
The stationary wavefunction ψ_{E}(ri) of the particle
satisfies the timeindependent Schrodinger equation:
H_{i}ψ_{Ei}(ri) = E_{i} ψ_{Ei}(ri)
We have finally:
ψ_{E}(r1, r2, ..., r_{N}) = Π(i) ψ_{Ei}(ri) =
ψ_{E1}(r1) ψ_{E2}(r2) ... ψ_{EN}(rN)
i from 1 to N
and
E = Σ(i) E_{i} = E_{1} + E_{2} + ... + E_{N}
i from 1 to N
ψ_{E}(r1, r2, ..., r_{N}) =
ψ_{E1}(r1) ψ_{E2}(r2) ... ψ_{EN}(rN)
E = Σ(i) E_{i} = E_{1} + E_{2} + ... + E_{N}
i from 1 to N
For non identical and noninteracting particles, the wave function of the whole system is
the product of the wave functions of the component particles; and the energy
of the whole system is the sum of the energies of the component particles.

