Schrodinger equation of particle in box

In summary, the Schrodinger Equation for a particle in a 1-dimensional box has two versions, one for the time-independent behavior and one for the time-dependent behavior. The solution to the time-dependent equation involves the term e-iEt/hbar. The general solution to the free Schrodinger Equation contains both sin and cos terms, but for the particle in a box, only sin terms are allowed due to the boundary conditions. This leads to a discrete set of particular solutions, explaining the quantized energy levels in systems such as atoms.
  • #1
R-ckay
4
0
hi all please help me... I'm learning schrodinger equation of a particle in a 1-dimensional box. I read a quantum mechanics book written by A. C Phillips. the wavefuction is

ψ (x,t)= N sin (kx) e-iEt/hbar

but when I compared to what I read from a modern physics book written by Beisser. the wavefunction is:

d2ψ/dx2 + 2m/hbar E ψ
=0

and the solution is:

ψ=A sin ((√2mE)/hbar)x + B cos ((√2mE)/hbar)

and we only use:

ψ =N sin (kx) only,

it is different. and I got trouble when I tried to find momentum probability. because it is different. which one is true?

my question is:
1.where "e-iEt/hbar" comes from?
2. how to get the solution "
ψ=A sin ((√2mE)/hbar)x + B cos ((√2mE)/hbar)" from
d2ψ/dx2 + 2m/hbar E ψ
= 0

thank you very much for help.. :)
 
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  • #2
#1: The sin(kx) is a solution to the time-independent Schrodinger Equation. This means that it only tells you the spatial behavior at the wave at one point in time. To find the time-dependence of the equation, you use the time-dependent version of the Schrodinger Equation, which says:
[tex]i\hbar\frac{d}{dt}\Psi(t) = \hat{H}\Psi(t)[/tex]
If [itex]\Psi(x)[/itex] is an eigenstate of [itex]\hat{H}[/itex], then [itex]\hat{H}\Psi(x)=E\Psi(x)[/itex], so
[tex]i\hbar\frac{d}{dt}\Psi(t,x) = \hat{H}\Psi(t,x) = E\Psi(t,x)[/tex]
The solution to this differential equation (which you can check by substitution) is
[tex]\Psi(t,x) = e^{-iEt/\hbar}\Psi(0,x)[/tex]

#2: It's a differential equation, so the answer to how to find the solutions is basically "any way you can". :smile: In this case, you can make a guess about the form that solutions must take (sinusoidal), and then substitute them into the original equation to figure out whether your guess is right.

The second equation is correct that both sin and cos terms appear in the general solution to the free Schrodinger Equation. However, the first equation is correct that solutions to the equation in an infinite potential well only contain some of those terms. The reason that only sin terms appear in the solution to the particle in a box is that you also have to remember the boundary conditions. Since the box is infinite at x=0 and x=L, the wavefunction must be zero outside of that range, i.e. [itex]\Psi(0) = 0[/itex] and [itex]\Psi(L) = 0[/itex]. This means that none of the cos terms will work (they're all nonzero at [itex]x=0[/itex]), and of the sin terms, the only ones which work are the ones which are also zero at [itex]x=L[/itex]. This means you're limited to [itex]\Psi(x)=sin(2\pi n x/L)[/itex], where [itex]n[/itex] is an integer.

In this way, the continuous space of general solutions gets reduced to a discrete (or "quantized") set of particular solutions which obey the boundary conditions. This happens any time you have a potential well like this, and is the reason that there are discrete energy levels in systems such as atoms. It's also where "quantum mechanics" gets its name.
 
Last edited:

Related to Schrodinger equation of particle in box

What is the Schrodinger equation of a particle in a box?

The Schrodinger equation of a particle in a box is a mathematical equation that describes the behavior of a quantum particle confined to a finite region, known as a "box". It takes into account both the particle's position and its wave function, and allows for the prediction of the particle's energy and probability of being in a certain location within the box.

What is the significance of the "box" in the Schrodinger equation?

The "box" in the Schrodinger equation represents the boundaries of the region in which the particle is confined. This can be a physical box, but it can also be any finite region or potential well that restricts the particle's movement. The size and shape of the box can greatly affect the behavior of the particle, making it an important factor in the equation.

What is the relationship between the Schrodinger equation and quantum mechanics?

The Schrodinger equation is a fundamental equation in quantum mechanics, which is the branch of physics that deals with the behavior of particles on a very small scale. It describes the quantum state of a particle and how it changes over time, and has been instrumental in understanding the properties and behavior of subatomic particles.

What are the boundary conditions for solving the Schrodinger equation of a particle in a box?

The boundary conditions for a particle in a box are that the wave function must be continuous and finite at the boundaries of the box, and must also approach zero as the particle approaches the boundaries. These conditions help to determine the allowed energy states and the probability of finding the particle in different locations within the box.

How is the Schrodinger equation of a particle in a box solved?

The Schrodinger equation is typically solved using mathematical techniques such as separation of variables, which involves breaking down the equation into simpler parts that can be solved individually. The solutions to the equation are known as eigenfunctions and correspond to different energy states of the particle. These solutions can then be combined to determine the overall wave function and the probability of finding the particle in a specific location within the box.

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