One dimensional potential step

The plus/minus sign describes a particle moving in the positive/negative x direction. The value of the C± complex constants is undetermined. This is not a physical wave function, because it describes an electron which is present everywhere in space and has no uncertainty in its momentum. But (an infinite number of) such waves may be used to construct wave packets, which may describe real electrons. The value of E is not restricted to discreet values: the energy spectrum is continuous. The phase velocity of this wave

vph = ω k = E

~k = ~k

2m_e (3.5.10)

depends on the value of kwhich means that the relative phases of waves with different wave numbers k in a wave packet change over time even in vacuum.

This will lead to the spread of the wave packet over time (See Appendix22.2).

3.5.2 One dimensional potential step

An electron is moving in the following potential (Fig. 3.8:

V(x) =

(0 when x <0 V₀ when x≥0

Figure 3.8: Electron in a one dimensional potential step. The wave function is displayed for Etot < V0

Determine the wave function for an electron moving from the left to the right.

The total electron energyE can be either smaller or larger thanV0. Solution Because the potential divide the space (the x axis in our case) to two distinct parts, both with a constant potential, the solution of the time independent Schr¨odinger equation in our case is best calculated by solving two equations: one for x <0:

− ~² 2m_e

d²ϕ₁

d x² =Eϕ₁ and an one for x≥0:

− ~² 2m_e

d²ϕ_I

d x² +V₀ϕ₂ =Eϕ_I, from which

− ~² 2m_e

d²ϕII

d x² = (E −V₀)ϕ_II,

and connect the two solutions obtained using suitable boundary conditions. Here m_e is the electron mass.

Both equations have the same structure:

d²ϕ

d x² =−k²ϕ(x) with

k² =





 2meE

~² x <0 2m_e(E −V₀)

~² x≥0

which have solutions in the form of:

ϕ(x) = const· e^{±i k x}

The general solution of both equations is the linear combination of these, where we must distinguish between the values of k in the two regions. Instead of using subscripts we will denote k_I with k and k_II with q

ϕ_I(x) =A·e^{i k x}+B·e^{−i k x} ϕ_II(x) =C·e^{i q x}+D·e^{−i q x}

Here A and C are the amplitudes of the wave traveling in the pos-itive x direction, while B and D are the corresponding amplitudes for waves moving in the opposite direction.

Because an electron cannot be divided to two “half-electrons”, the wave function must be continuous. This is the first boundary con-dition.

And because the resulting wave function must be the solution of a single equation containing a second derivative for the whole space, the second boundary condition is thecontinuity of the first deriva-tive of the wave function at x= 0:

Our electron originally arrives to the x= 0 boundary from the left traveling in the positive x direction. This means that the amplitude A 6= 0 for the wave traveling right in region ’I’. Part of the wave function may be reflected back from the potential step into region

’I’ (B 6= 0) and part of it may enter the region of the higher potential (C 6= 0). But there will be no part traveling backwards there, therefore D= 0.

Substituting the wave functions into the boundary conditions we get the following equalities:

A+B =C i k(A−B) = i q C

i.e. 2 equations for the 3 unknowns. This means that we can set the value of one of the unknown parameters arbitrarily and determine the others depending on its value²⁰. In our case let us select the value A = 1. With this selection

B = k−q

Up to this point we did not distinguish between the two cases when E > V₀ and when E < V₀.

When E > V0, i.e. the kinetic energy of the incoming particle is larger than the potential step both k and q are real. The part of the wave function which enters region ’II’ is

ϕ₂(x)≡C·e^{−i q x}

which is a wave traveling in the positive x direction with constant amplitude and constant |ϕ_II|² =|C|² probability density.

If the electron was a classical particle, whose movement is governed by the laws of classical mechanics it could never turn around, it would always move in the positive x direction. In quantum me-chanics however there is always some possibility that the electron is reflected back from the boundary, because if V₀ >0 then B 6= 0.

When E < V₀,i.e. the kinetic energy of the incoming particle is larger than the potential step then k is still real but q is imagi-nary: q = i

√

2me(V0−E)

~ , this makes B and C complex. Part of the wave function still reaches into region ’II’ but it is an exponentially decreasing function:

ϕ_II(x) =C·e^i·i·|q|x =C·e^−·|q|x

In classical physics if the total energy of the particle is smaller than the potential energy in some region of space the particle is

20Because the wave arriving from the left is infinite it can not be normalized.

always reflected back from the boundary and cannot enter the re-gion where E < V₀. This reflection always occurs in quantum me-chanics too, but the wave function will not be 0 inside region ’II’.

The probability density decreases exponentially from the bound-ary: |ϕ₂(x)|² =|C|²e^−|q|x. The penetration depth δ_P is the distance where the probability density falls to 1/e (about 37%) of its value at the boundary, i.e.

δ_P = ~

p2m_e(V₀− E)

I.e. the higher is the potential the smaller is the penetration depth.

We learned from this calculation that

Important 3.5.5. The solution of the Schr¨odinger equaton (the wave function) must be a finite valued, continuous and continuously differentiable function. This is true even for V(x) potentials that have an abrupt, but finite jump.

The potential we used in this example is of course an idealization. In realistic cases (see the next section)the potential raises from 0 to a constant value not abruptly but in some

∆x=sdistance. In classical mechanics you may imagine a ramp connecting the ground with a raised platform. The potential energy of a classical object sliding or rolling on this ramp has a potential energy of this shape. In quantum mechanics this potential can be realized by a homogeneous electric field connecting two halves of space with different potential.

But do not forget that the particle is moving along the x axis and not along the potential curve!