[Quantum mechanics] Step barrier |R|^2 and |T|^2

In summary, the conversation is about finding expressions for |R|^2 and |T|^2, as well as discussing the confusion about using either T = 1 + R or T = 1 - R in the calculations. There is also a mention of the probability interpretation and the normalization of wavefunctions. The equations for R and T derived in the conversation are assumed to be correct, but it is important to note that T = 1 - R would violate the Schrodinger equation at x = 0. The comparison of |R|^2 and |T|^2 as probabilities may not make sense in this context.
  • #1
Aaron7
14
0

Homework Statement



There is a step barrier at x=0, V_0 > E
I am given:
ψi = e^i(kx−ωt) --->
ψr = R e^i(−kx−ωt) <---

ψt = T e^(−αx−iωt) --->

Part of question I am confused about: State the two boundary conditions satisfied by the wave function at x = 0 and hence find expressions for |R|^2 and |T|^2

Homework Equations


N/A


The Attempt at a Solution


I have already worked out an equation for alpha in the previous part.

ψ1 = ψi + ψr
ψ2 = ψt

I started to apply the conditions ψ1(0) = ψ2(0) to get 1 + R = T
and d/dx ψ1(0) = d/dx ψ2(0) to get ik - iRk = -αT

I solved the above to get R = (ik +αT) / ik with T = 1 + R etc
I understand that |R|^2 = R x R* and |T|^2 = T x T*
However I am told that T = 1 - R since these are probability coefficients. Do I solve with T = 1-R (probability coefficients) or T = 1+R (using ψ1(0) = ψ2(0))? I am confused which one to use and why.

Many thanks.
 
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  • #2
Aaron7 said:
ψi = e^i(kx−ωt) --->
ψr = R e^i(−kx−ωt) <---

ψt = T e^(−αx−iωt) --->

ψ1 = ψi + ψr
ψ2 = ψt

I started to apply the conditions ψ1(0) = ψ2(0) to get 1 + R = T
and d/dx ψ1(0) = d/dx ψ2(0) to get ik - iRk = -αT

However I am told that T = 1 - R since these are probability coefficients.
Well, ask yourself: should you expect the wavefunction to violate the Schrodinger equation at x=0, and if so, why?

The probability interpretation only makes since with respect to a normalization, usually the L2 norm for wavefunctions. Unfortunately, the friendly plane wave that makes naive calculations so simple wreaks havoc on the L2 norm. (Plane waves for free particles actually live in a different kind of quantum space than normalizable/quantized wavefunctions.) I don't know what you mean by calling R and T "probability coefficients", but I believe that your equations for R and T (the ones that you derived) are correct (assuming that the wavefunction obeys the Schrodinger equation, which T = 1 - R would actually violate at x = 0).

I think it doesn't even make sense to compare |R|2 to |T|2 as probabilities, because I think that the probability of transmission is zero, but I will have to think back to my QM (and that is a long way back).
 

FAQ: [Quantum mechanics] Step barrier |R|^2 and |T|^2

What is the significance of the quantities |R|^2 and |T|^2 in quantum mechanics step barrier?

The quantities |R|^2 and |T|^2 represent the probabilities of reflection and transmission, respectively, for a particle encountering a step barrier in quantum mechanics. They are calculated using the wave function of the particle and the potential energy of the barrier.

How do the values of |R|^2 and |T|^2 change with the height and width of the step barrier?

The values of |R|^2 and |T|^2 are dependent on the height and width of the step barrier. As the height increases, the probability of reflection also increases, while the probability of transmission decreases. Similarly, as the width increases, the probability of reflection decreases, while the probability of transmission increases.

What is the relationship between |R|^2 and |T|^2 in quantum mechanics step barrier?

In quantum mechanics, the sum of |R|^2 and |T|^2 is always equal to 1. This is known as the conservation of probability, which states that the total probability of all possible outcomes must equal 1.

Can |R|^2 and |T|^2 be greater than 1 in quantum mechanics step barrier?

No, |R|^2 and |T|^2 cannot be greater than 1 in quantum mechanics step barrier. As mentioned before, the sum of these quantities must always be equal to 1, and probabilities cannot exceed 1.

How do |R|^2 and |T|^2 change with the energy of the particle in quantum mechanics step barrier?

The values of |R|^2 and |T|^2 are also dependent on the energy of the particle. As the energy increases, the probabilities of both reflection and transmission also increase. However, the exact relationship between energy and these probabilities is determined by the specific potential energy of the barrier.

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