The document Tunneling through a Barrier(Square and Step Barriers) - The Schrodinger Equation, Quantum Mechanics, Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.

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**The square barrier:**

**Behaviour of a classical ball rolling towards a hill (potential barrier):**

If the ball has energy E less than the potential energy barrier (U=mgy), then it will not get over the hill.

The other side of the hill is a classically forbidden region.

**Behaviour of a quantum particle at a potential barrier **

Solving the TISE for the square barrier problem yields a peculiar result:

If the quantum particle has energy E less than the potential energy barrier U, there is still a non-zero probability of ﬁnding the particle classically forbidden region !

This phenomenon is called tunneling.

**Behaviour of a quantum particle at a potential barrier**

To the left of the barrier (region I), U=0 Solutions are free particle plane waves:

The ﬁrst term is the incident wave moving to the right The second term is the reﬂected wave moving to the left.

**Reﬂection coefﬁcient:**

**Behaviour of a quantum particle at a potential barrier**

To the right of the barrier (region III), U=0. Solutions are free particle plane waves:

This is the transmitted wave moving to the right

**Transmission coefﬁcient**:

**Behaviour of a quantum particle at a potential barrier**

In the barrier region (region II), the TISE is

**Solutions are**

**Behaviour of a quantum particle at a potential barrier **

At x=0, region I wave function = region II wave function:

At x=L, region II wave function = region III wave function:

**Behaviour of a quantum particle at a potential barrier **

At x=0, dϕ/dx in region I = dϕ/dx in region II:

At x=L, dϕ/dx in region II = dϕ/dx in region III :

**Behaviour of a quantum particle at a potential barrier **

Solving the 4 equations, we get

For low energies and wide barriers,

For some energies, T=1, so the wave function is fully transmitted (transmission resonances).

This occurs due to wave interference, so that the reﬂected wave function is completely suppressed.

**The step barrier:**

To the left of the barrier (region I), U=0.

Solutions are free particle plane waves:

**Inside Step:**

U = V_{o}

**The step barrier:**

R(reflection) + T(transmission) = 1

Reflection occurs at a barrier (R ≠ 0), regardless if it is step-down or step-up.

R depends on the wave vector difference (k_{1} - k_{2}) (or energy difference), but not on which is larger.

Classically, R = 0 for energy E larger than potential barrier (V_{o}).

A free particle of mass m, wave number k_{1} , and energy E = 2V_{o} is traveling to the right. At x = 0, the potential jumps from zero to –V_{o} and remains at this value for positive x. Find the wavenumber k_{2} in the region x > 0 in terms of k_{1} and V_{o}. Find the reflection and transmission coefficients R and T.

Sketch the wave function ψ(x) corresponding to a particle with energy E in the potential well shown below. Explain how and why the wavelengths and amplitudes of ψ(x) are different in regions 1 and 2.

ψ(x) oscillates inside the potential well because E > V(x), and decays exponentially outside the well because E < V(x).

The frequency of ψ(x) is higher in Region 1 vs. Region 2 because the kinetic energy is higher [E_{k} = E - V(x)].

The amplitude of ψ(x) is lower in Region 1 because its higher E_{k} gives a higher velocity, and the particle therefore spends less time in that region.

Sketch the wave function ψ(x) corresponding to a particle with energy E in the potential shown below. Explain how and why the wavelengths and amplitudes of ψ(x) are different in regions 1 and 3.

ψ(x) oscillates in regions 1 and 3 because E > V(x), and decays exponentially in region 2 because E < V(x).

Frequency of ψ(x) is higher in Region 1 vs. 3 because kinetic energy is higher there.

Amplitude of ψ(x) in Regions 1 and 3 depends on the initial location of the wave packet. If we assume a bound particle in Region 1, then the amplitude is higher there and decays into Region 3 (case shown above).

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