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


Quantum Mechanics Operators
Operators in Quantum Mechanics are useful tools to do
the related mathematics calculations more rapidly. Quantum
Mechanics is just Quantum Mathematics operating all the time
on the wave function ψ(r,t).
1. operation
an operation is an action that produces a new value
from one or more input values.
2. operator
an operator is a symbol or function that represents a
mathematical operation.
3. Selfadjoint operator
a selfadjoint operator is an operator that is its own adjoint
4. Adjoint operator
adjoint = Conjugate transpose
5. Conjugate Operator
The conjugate of the complex matrix A, written as A*, is
obtained by taking the complex conjugate of each entry.
6. Transpose operator
If A is an ij matrix, then the (i,j) element of itf
transpose AT is the (j,i) element of A.
[A^{T}]ij = [A]ji
7. Conjugate transpose operator
The conjugate transpose of the complex matrix A, written as A*, is
obtained by taking the transpose of A and the complex conjugate of
each entry.
8. Hermitian operator
If A = its adjoint, then A is called Hermitian
In quantum mechanics an observable is a Hermitian operator on
the physical Hilbert space.
9. Linear operator
A linear map, or linear operator is a function between two
vector spaces that obeys the operations of vector addition
and scalar multiplication; that is:
If A and B are vector spaces over the same field C, and a function
f (or oerator f) : A → B is linear if for any two
vectors x and y in A and any scalar a in C, we have:
f(x + y) = f(x) + f(y),a nd
f(α x) = α f(s)
Or, in general manner:f(Σ α_{n} x_{n}) = Σ α_{n} f(x_{n} )
10. Observable
Any measurable quantity for which we can calculate the
expectation value is called a physical observable.
11. Observable operators
We take the simplest case where the particle is free, that is
V(r) = 0. Thereore: ψ(x,t) = exp{i(kx  ωt)}
1. Linear momenum Operator
∂ ψ(x,t) /∂x = ik ψ(x,t)
With p = ℏ k, we have:
∂ ψ(x,t) /∂x = (ip/ℏ) ψ(x,t)
then:
p =  i ℏ) ∂ /∂x
This is the operator associated to the physical observable
linear momentum P.
Linear momenum Operator:
p =  i ℏ ∂ /∂x
2. Energy Operator
Now, the time derivative of the wave function is:
∂ ψ(x,t) /∂t =  i ω ψ(x,t)
With E = ℏ ω, we have:
∂ ψ(x,t) /∂t = ( iE/ℏ) ψ(x,t)
then:
E = i ℏ ∂ /∂t
This is the operator associated to the physical observable
energy.
Energy Operator:
E = i ℏ ∂ /∂t
3. Timedependent Schrodinger equation
Using operators, we can verify the timedependent
Schrodinger equation:
p =  i ℏ ∂ /∂x
p^{2} =  ℏ^{2} ∂^{2} /∂x^{2}
KE = p^{2}/2m =  (ℏ^{2}/2m) ∂^{2} /∂x^{2}
If we use the conservation of energy, that is KE + PE = E,
and take the simplest case where the particle is free (PE = 0),
we have KE = E, and then :
 (ℏ^{2}/2m) ∂^{2} /∂x^{2} = i ℏ ∂ /∂t . That is the timedependent
Schrodinger equation.
4. General case
The Schrödinger timeindependant equation:
 (ℏ ^{2}/2m)∂^{2}ψ(r)/∂r^{2} + V(r) ψ(r) = E ψ(r)
It gives the total energy operator Hamiltonian E:
E =  (ℏ ^{2}/2m)∂^{2}/∂r^{2} + V(r)
The Schrödinger timedependant equation:
iℏ ∂ψ(r,t)/∂t =  (ℏ ^{2}/2m)∂^{2}ψ(r,t)/∂r^{2} + V(r) ψ(r,t)
It gives the Hamiltonian E:
E = iℏ ∂/∂t
The kinetic energy operator KE is:
KE =  (ℏ ^{2}/2m)∂^{2}/∂r^{2}
If KE = P^{2} /2m =  (ℏ ^{2}/2m)∂^{2}/∂r^{2}, then:
P =  i ℏ∂/∂r
Remark ∂/∂r (∂/∂r) =  ∂^{2}/∂r^{2} when
acting on ψ(r), because ψ(r) = ψ( ω t  kr) .
12. Commutator
By definition, the commutator of two operators
A and B is : [A,B] = AB  BA
The two operators A and B commute if [A,B] = 0.
Examples:
1. The two integers 1 and 3 commute because [1,3] = 1x3  3x1 = 0.
2. The momentum and position do not.
Proof:
p =  i ℏ ∂ /∂x
[p, x] = px  xp
px ψ = p (x ψ) =  i ℏ ∂(x ψ) /∂x =  i ℏ (ψ + x ∂ψ/∂x )
And:
xp ψ = x p(ψ) = x ( i ℏ ∂ ψ/∂x )
Thus:
[p, x] ψ =  i ℏ (ψ + x ∂x (ψ) /∂x )  x ( i ℏ ∂ (ψ)/∂x ) =
iℏ ( ψ  x ∂x (ψ) /∂x + x ∂ (ψ)/∂x ) =  i ℏ ψ
Therefore: [P,X] =  i ℏ
A manner to show that we can not know two physical values
(here position and momentum) at the same time if they do not
commute.

