We begin by restricting to the special case where the potential energy V is not a function of time but depending only on x. This is a true, if the system experiences have no time - dependent external forces. The time - dependent of the Schrodinger equation reads,
......... (1)
Now we restrict looking for that solution of equation (1) that can be written as a product of the function of time and a function of x:
(capital) ψ (x, t) =f (t) small φ(x)........ (2)
The capital ψ is used for the time - dependent wave function and lowercase φ used for the factor that depends only one the coordinate x. The states corresponding to the wave function of the form equation (2) possess certain properties that making them of great intervest. Taking partial derivatives of Equation (2)
Substitution into equation... (1) gives
Where we can divided by fψ .in generally we can expect the quantity to which each side.. (3) is equal to be a certain function of x and t. However, the right side of.. (3) doesn't depend on t, so that function to which each side of.. (1) is equal must be independent of t. The left side.. (1) to E, we getting
df(t) /f(t) =- iE/ħ dt
Integrating take both sides of the this equation with respect to t, and we have
Inf(t) =-iEt/ħ +C
Where C is an arbitrary constant value of integration, Hence
f(t) =ₑᶜₑ −ᵢEt/ħ =Ae⁻ᵢEt/ħ
Where the arbitrary constant A has replaced eᶜ. Since A can be induced as a factor in the function ψ (x) that multiplies f (t) in equation (2), A can be be omitted from f(t) Thus
f(t) = e⁻ᵢEt/ħ
Equating the right side.. (3) to E, we have
Equation... (4)
The equation (4) is the time - independent Schrodinger equation for a single particle of mass m moving in one dimensional.
The significance of the constant, E, and E occurs as [ E−V (x) ] in equation.. (4)
And E has the same dimensional as V, so E has the dimensions of energy. In fact, we can postulated that E is the energy of the system. Thus, for cases where the potential energy is a function of x only, there exists the wave function of the form
ψ(x, t) = e⁻ⁱEt/ħψ (x)........(5)
and this wave function corresponding to states of constant energy E.
The wave function in (5) is complex, but the quantity thats experimentally observed is the probability density lψ (x, t)l². Thus square of the absolutely value of a complex quantity is given by the product of the quantity with it's complex conjugate, the complex conjugate being formed by the replacing i with - i wherever it's occurs.
Iψl² =ψ*ψ......(6)
Where the star is denotes the complex conjunction. For the wave function equation (5)
......... (7)
In driving equation (7)an we can assuming that E is a real number, so E= E*.
Hence for states of the form equation (5) the probability density is given by lψ(x) l² and doesn't change with time. Such states are called stationery states. Since the physically significant quantity is lψ(x, t)l² and since for stationary states lψ(x, t)l² =l ψ(x)l² the function ψ (x) is often called the wave function. Although the complete wave function of a stationary state is obtained by multiplying ψ (x) by e⁻iEt/ħ. The term is stationary state is at rest.
Note the Schrodinger equation contains two unknowns. The allowed energies E and the allowed the wave function ψ.To solve for two unknowns, we need to impose additional conditions it's called boundary conditions. On ψ besides required that it's satisfy equation (4).The boundary conditions determine the allowed energies, since it's turn out that only certain values of E allow ψ to satisfy boundary conditions.
Very nice
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