# Tunneling Electrons

When considering an electron in a potential well, why is it possible for the electron to tunnel through a barrier of finite potential and not an infinite well.

I can see the equations that disprove it (textbook) but i can't see any REASONS persay.

Thanks.

malawi_glenn
Homework Helper
the reasons are due to the equations ;-)

The reasons for that a electron CAN tunnel are due to the equations, why are the equations not valid when they say when an electron CAN'T tunnel? That is my ultimate question to you.

Of course, infinity does not exists in reality either.

Nothing can beat infinity in a fight (except a stronger/better/faster infinity).

You can think of electrons getting through finite potential barriers as them borrowing enough energy to squeeze through without anyone noticing. For an infinite barrier, they would have to borrow an infinite amount of energy... thats no good.

Also an electron can exist simultaneously inside and outside the potential well. If the well is finite then the probability of detecting the electron outside is higher, but if the well is infinite then that probability of detecting the electron outside approaches zero.

malawi_glenn
Homework Helper
Izkelly, the electron don't borrow energy. The probability that it can be in a classical forbidden region are due to the Shrödinger equation and the requirment that the wavefunction is continous.

Waht, the OP's question was WHY is the probability zero when the well is infinite high.

if it doesn't "borrow" energy (uncertainty principle style) then it seems like it would have a negative kinetic energy inside the barrier... ?
My understanding (though very possibly incorrect) was what allowed it to have a finite probability density past the barrier was due to "borrowing" energy.

malawi_glenn
Homework Helper
if it doesn't "borrow" energy (uncertainty principle style) then it seems like it would have a negative kinetic energy inside the barrier... ?
My understanding (though very possibly incorrect) was what allowed it to have a finite probability density past the barrier was due to "borrowing" energy.

Where in the derivation of the Schrodinger equation and tunneling does borrowing energy comes in? (nowhere. I have derived that equation..)

From and old thread about a similar topic, posted ZapperZ, one of the mentors of this forum (and hence one of the most educated)

"As far as the "negative KE" thing, I can see how that might come about since we typically define the total energy E as E=PE + KE. Since in the barrier, E is less than PE, one would tend to think that KE is negative here. However, again, this may not be the right picture. Remember that all we care about is the wave vector "k" in the wavefunction. For a free particle in 1D, it is just exp(ikx). k here is defined as being proportional to sqrt(E-PE). Depending on the sign of E-PE, one and re-adjust k to be either real or imaginary, resulting in either a propagating wave, or a decaying wave. This is all that matters. It makes it an added complication to try and equate this to "KE", which in itself is not well-defined in the barrier (remember, v or p is again an observable that we don't know of in the barrier)."

ZapperZ
Staff Emeritus
When considering an electron in a potential well, why is it possible for the electron to tunnel through a barrier of finite potential and not an infinite well.

I can see the equations that disprove it (textbook) but i can't see any REASONS persay.

Thanks.

You need to be a bit careful here. The nature of the infinite barrier makes a difference.

Whenever you try to figure out something like this, it is always useful to write down the schrodinger equation for each region, and then do the boundary conditions. You'll see the reason mathematically why the wavefunction "dies down" to zero at the boundary for the infinite barrier.

However, this is NOT TRUE for a delta function barrier, which has an infinite height but infinitely small width. Here, if you do the math carefully, you'll find that across the delta function barrier, the wavefunction has a "derivative jump", i.e. while the wavefunction is continuous, its derivative across the delta function barrier isn't.

Zz.

Where in the derivation of the Schrodinger equation and tunneling does borrowing energy comes in? (nowhere. I have derived that equation..)

A little off-topic, but... Is the Schrodinger equation derived? I thought it was basically fundamental (with the exception of the Dirac equation), and chosen as an axiom because its results are compatible with experimental data, not because it results from some other more fundamental mathematics.

If it is derived, in what book may I find its derivation?

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the "derivation" of the Schroedinger equation is an exploitation of the ANALOGY of mechanics to QM, and is most certainly not a derivation in that its premises are strongly a posteriori. Ironically, that "derivation" is founded on E = K + U, exactly the source of my confusion that glenn clarified.

Ahh. I figured that I might have been taking the word "derivation" too seriously, but I felt the need to clarify it for sure.

malawi_glenn
Homework Helper
Ahh. I figured that I might have been taking the word "derivation" too seriously, but I felt the need to clarify it for sure.

Well you can derive the Shcrödinger equation from the postulates of QM. Remember that QM has certain postulates that are the Quantum mechanical commutators, which is the classical Poission brackets, but times (-i/hbar). From this, you can derive the shrodinger equation.

Just as E=mc^2 can be derived from the postulates of Special relativity.

Also, the Dirac equation is indeed derived.

lzkelley: You dont sum the K and U to get E, you operate on the state. You 'count' different in QM.

An exercise: Find the ground state wave function of a system (an electron):

V(x) = + infinity for x < 0
V(x) = 0 for 0<x< a, where a is 3 nm
V(x) = 1eV for x> a

Find also the kinetic energy of the particle in a region x>a

The kinetic energy is given by applying the Kinetic Hamiltonian, $$\frac{-\hbar^2}{2m_e}\frac{d^2}{dx^2}$$

Just for fun, nothing else :)

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Defennder
Homework Helper
the reasons are due to the equations ;-)

The reasons for that a electron CAN tunnel are due to the equations, why are the equations not valid when they say when an electron CAN'T tunnel? That is my ultimate question to you.

Of course, infinity does not exists in reality either.
Which equations are we talking about?

malawi_glenn
Homework Helper
Which equations are we talking about?

I dont know, the Schrödinger equation and the wave functions that it produces?

"The postulates" of the QM are founded on the Schrodinger equation, not visa versa.
And i said its founded on E = K + U, founded means thats the basis. Just like QM is "founded" on Schrody, Heisi, and the Broge.

malawi_glenn
Homework Helper
"The postulates" of the QM are founded on the Schrodinger equation, not visa versa.
And i said its founded on E = K + U, founded means thats the basis. Just like QM is "founded" on Schrody, Heisi, and the Broge.

So the quantum mechanical commutators are postulated from a postulated / semi derived Schrodinger equation?

Where can I read that historical treatment?

malawi_glenn