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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. Self-adjoint operator
a self-adjoint 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. [AT]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 αn xn) = Σ αn f(xn )
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. Time-dependent Schrodinger equation

Using operators, we can verify the time-dependent Schrodinger equation: p = - i ℏ ∂ /∂x p2 = - ℏ22 /∂x2 KE = p2/2m = - (ℏ2/2m) ∂2 /∂x2 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 : - (ℏ2/2m) ∂2 /∂x2 = i ℏ ∂ /∂t . That is the time-dependent Schrodinger equation.

4. General case

The Schrödinger time-independant equation: - (ℏ 2/2m)∂2ψ(r)/∂r2 + V(r) ψ(r) = E ψ(r) It gives the total energy operator Hamiltonian E: E = - (ℏ 2/2m)∂2/∂r2 + V(r) The Schrödinger time-dependant equation: iℏ ∂ψ(r,t)/∂t = - (ℏ 2/2m)∂2ψ(r,t)/∂r2 + V(r) ψ(r,t) It gives the Hamiltonian E: E = iℏ ∂/∂t The kinetic energy operator KE is: KE = - (ℏ 2/2m)∂2/∂r2 If KE = P2 /2m = - (ℏ 2/2m)∂2/∂r2, then: P = - i ℏ∂/∂r Remark ∂/∂r (∂/∂r) = - ∂2/∂r2 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.

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