If V is only a function of x, then the equation's general solution can be nearly solved using the method of separation of variables.
Starting with the time dependent Schrödinger equation:
(iℏ)∂Ψ/∂t = -(ℏ2/2m)∂2Ψ/∂x2 + VΨ, where iis the imaginary number, ℏ is Plank's constant/2π, Ψ is the wave function, m is mass, x is position, t is time, and V is the potential and is only a function of x in this case.
Now we find solutions of Ψ that are products of functions of either variable, i.e.
Ψ(x,t) = ψ(x) f(t)
Taking the first partial of Ψ in the above equation with respect to t gives:
∂Ψ/∂t = ψ df/dt
Taking the second partial of Ψ in that same equation above with respect to x gives:
∂2Ψ/∂x2 = d2ψ/dx2 f
Substituting these ordinary derivatives into the time dependent Schrödinger equation gives:
(iℏψ)df/dt = -(ℏ2/2m)d2ψ/dx2f + Vψf
Dividing through by ψf gives:
(iℏ)(1/f)df/dt = -(ℏ2/2m)(1/ψ)d2ψ/dx2 + V
This makes the left side of the equation a function of tonly, and the right a function of x only; meaning both sides have to be a constant for the equation to hold (I'm not going to prove why this is so, because if you've understood what I've written up to this point, you're probably familiar with the separation of variables technique).
That allows us to make two separate ordinary differential equations:
1) (1/f)df/dt = -(iE/ℏ)
2) -(ℏ2/2m)d2ψ/dx2 + Vψ = Eψ, where E is the constant that both separated differential equations must equal to (I did a little bit of algebra to these equations also). This is known as the time independent Schrödinger equation.
To solve 1) just multiply both sides of the equation by dt and integrate:
ln(f) = -(iEt/ℏ) + C. Exponentiating and, since C will be absorbed later, removing C:
f(t) = e-(iEt/ℏ)
Now, we have gone as far as we can until the potential, V(x), is specified. That leaves us with the general solution to the time dependent Schrödinger equation being:
Ψ(x,t) = ψ(x) e-(iEt/ℏ)
There are many solvable examples of ψ(x) for specific V(x). I've linked some below. Also, if V is a function of both x and t, there are methods to find solutions, such as perturbation theory and adiabatic approximation.
The time-independent Schr
The basic definition of speed is: speed = distance / time Solve this equation for distance, or solve it for time, to get two additional versions of the equation.
The equation 31 time 744 equals 23,064. All you have to do is add up 744 31 times.
You can solve for a one-time constant by using the formula t = RC. Read the math problem you are given carefully to determine what values to plug into the equation.
You forgot to include the equation. Just replace y with 5, every time it occurs, then solve the remaining equation for x.
The time-independent Schr
The time dependent equation is more general. The time independent equation only applies to standing waves.
Being a physicist I do not know too much about the applications. But in general the time dependent Schrodinger Equation tells us how a quantum state evolves in time. I believe this might be applicable to things like flash/thumb drives, and computers in general.
The basic definition of speed is: speed = distance / time Solve this equation for distance, or solve it for time, to get two additional versions of the equation.
You should substitute your solution in the equation. If the solution is correct you will receive equality. Otherwise your solution is wrong.
The equation 31 time 744 equals 23,064. All you have to do is add up 744 31 times.
Use the equation, speed = distance / time, substitute in the given information from the problem and solve it.
You can solve for a one-time constant by using the formula t = RC. Read the math problem you are given carefully to determine what values to plug into the equation.
You forgot to include the equation. Just replace y with 5, every time it occurs, then solve the remaining equation for x.
Erwin Schrodinger invented the model of the atom based on research done by scientists such as Niels Bohr.
vf=vi+at equation can be solved by substituting the letters in the equation with there actual values where vf is the finall velocity, vi is the initial velocity, a is the acceleration and t is the time.
Because if you put there dependence on time for example you will not able to solve equation for a reasonable period of time. But sometimes you have to use potentials with time and coordinate dependence, for instance, when you work with magnetic fields.