How Do You Determine the Wave Function Across a Delta Potential Barrier?

[arierreF]

Homework Statement

V(x) = \begin{cases}<br /> \infty &amp; , &amp; x&lt;0 \\<br /> -g*\delta (x-a) &amp; , &amp; x&gt;0<br /> \end{cases}<br /> \text{ }Write the wave function in the left side of x=a and right side of x=a (BOUND STATE)

The Attempt at a Solution

In the right side o x=a i would say that it is \psi (x)=\text{Be}^{-\text{kx}}.

But in the left side, i am having problems figuring out what is the wave function.

Is it \psi (x)=\text{Be}^{\text{kx}} in the even case and \psi (x)=-\text{Be}^{\text{kx}} in the odd case??

Or because it is between two potential, it is \psi (x)=A *\text{Cos}[\text{px}] even case.

\psi (x)=A *\text{Sin}[\text{px}] odd case.

 

[dextercioby]

Who's k ?

 

[arierreF]

k is equal to \sqrt{ \frac{2 *m |E|}{\hbar^{2}}}

\frac{d^2\psi }{dx^2}= \frac{2*m*|E|}{\hbar{}^{\wedge}2}\psi (BOUND STATE)

 

[arierreF]

any help?

 

[TSny]

Suppose you start with a simpler case where there is no delt-function potential, so that
V(x) = \begin{cases}<br /> \infty &amp; , &amp; x&lt;0 \\<br /> 0 &amp; , &amp; x&gt;0<br /> \end{cases}<br /> \text{ }

What would be the form of the wavefunction for x > 0 that satisfies the boundary condition at x = 0?

 

[arierreF]

At x>0 Potential is 0

\frac{-h{}^{\wedge}2}{2m}\frac{d^2\psi }{dx{}^{\wedge}2} = \text{E$\psi $}k= \sqrt{\frac{2m}{h{}^{\wedge}2}E}
bound state, k is real.

\psi = \text{Ce}^{-\text{kx}} +\text{De}^{\text{kx}}\text{ }= \text{Ce}^{-\text{kx}}

e^kx blows up in infinity.
Other possibility is the particle coming from the right without amplitude , be totally reflected in the infinity barrer at x=0, then it goes to plus infinity with amplitude.

 

[TSny]

k= \sqrt{\frac{2m}{h{}^{\wedge}2}E}
bound state, k is real.

\psi = \text{Ce}^{-\text{kx}} +\text{De}^{\text{kx}}\text{ }= \text{Ce}^{-\text{kx}}

e^kx blows up in infinity.

You can check that $\small e^{kx}$ and $\small e^{-kx}$ do not satisfy the free-particle Schrodinger equation for positive E.

Shouldn't a free particle have an oscillatory wave function?

 

[arierreF]

But in the bound states E<0.

So is satisfied e^kx and e^-kx.

I thought that only when E>0, the solutions are e^ikx and e^-ikx, i.e, particle has oscillatory motion.

 

[TSny]

Yes, You are right. I overlooked that you are asked to find bound states. Without the delta potential there would not be bound states. Sorry about that. So, let's go back to the original problem with the delta-potential in place.

For 0< x < a, can you find a superposition of $e^{kx}$ and $e^{-kx}$ that will satisfy the boundary condition at x = 0?

 

[arierreF]

maybe a 2 cosh = e^kx + e^-kx (even case)

 

[TSny]

maybe a 2 cosh = e^kx + e^-kx (even case)

What should ψ(0) equal?

 

[arierreF]

I guess that should be zero. As in the infinite box.

My problem is not knowing the "effect" of the infinite potential.

I know that is left side of delta x<a, we have Psi(x) = Be^kx

 

[TSny]

Both e^kx and e^-kx satisfy the Schrodinger equation in the region 0<x<a. So, any linear combination of those two functions also satisfies the Schrodinger equation in this region.

You are correct that the infinite potential at x = 0 implies the boundary condition ψ(0) = 0. What linear combination of e^kx and e^-kx satisfies this boundary condition?

 

[arierreF]

Both e^kx and e^-kx satisfy the Schrodinger equation in the region 0<x<a. So, any linear combination of those two functions also satisfies the Schrodinger equation in this region.

You are correct that the infinite potential at x = 0 implies the boundary condition ψ(0) = 0. What linear combination of e^kx and e^-kx satisfies this boundary condition?

well, its ψ(0) = 0 => e^kx - e^-kx

it is a hyperbolic sin.

 

[TSny]

well, its ψ(0) = 0 => e^kx - e^-kx

it is a hyperbolic sin.

Yes. And any multiple of sinh(kx) will also satisfy the boundary condition at x = 0 and also satisfy the Schrodinger equation in region I where 0<x<a.

So, the form of the solution in region I is ψ_I(x) = Asinh(kx)

You have already identified the form of the solution for region II (x>a): ψ_II(x) = Be^-kx.

To relate A to B, we need another boundary condition. What can you say about the wavefunction at x = a?

 

[arierreF]

Yes. And any multiple of sinh(kx) will also satisfy the boundary condition at x = 0 and also satisfy the Schrodinger equation in region I where 0<x<a.

So, the form of the solution in region I is ψ_I(x) = Asinh(kx)

You have already identified the form of the solution for region II (x>a): ψ_II(x) = Be^-kx.

To relate A to B, we need another boundary condition. What can you say about the wavefunction at x = a?

ψ_II(a) = ψ_I(a)

ψ'_I(a) - ψ'_II(a) = -2mg/\hbar * ψ_I(a)Its going to give a transcendent equation.

 

[TSny]

ψ_II(a) = ψ_I(a)

Yes. This equation can be used to relate A and B.

Would normalization provide another relation between A and B. If so, then A and B are essentially determined at this point.

ψ'_I(a) - ψ'_II(a) = -2mg/\hbar * ψ_I(a)

Its going to give a transcendent equation.

I don't quite get the same sign and numerical factor for the right hand side of this equation.
Yes, this will yield a transcendental equation. Since A and B are already determined, what important information will this equation provide?

 

[arierreF]

I don't quite get the same sign and numerical factor for the right hand side of this equation.
Yes, this will yield a transcendental equation. Since A and B are already determined, what important information will this equation provide?

I checked the equation and i get the negative sign in the RHS of the equation. V = -g δ(x-a)
is going to the right side, and then we multiple it by the -h^2 /2m. The transcendental equation is going to provide the values of the energy eigenvalues.

Sorry my english btw.

 

[TSny]

I checked the equation and i get the negative sign in the RHS of the equation. V = -g δ(x-a)
is going to the right side, and then we multiple it by the -h^2 /2m.

Should the left hand side be
ψ'_I(a) - ψ'_II(a)
or
ψ'_II(a) - ψ'_I(a) ?

The transcendental equation is going to provide the values of the energy eigenvalues.

Yes. Good.

 

[arierreF]

Should the left hand side be
ψ'_I(a) - ψ'_II(a)
or
ψ'_II(a) - ψ'_I(a) ?Yes. Good.

You are right.

I was thinking that the zone II was the left zone.

Thanks for all the help!