# Exploring the Time-Energy Uncertainty Relation to Solve a Schrodinger Equation

• Wannabe Physicist
In summary, the conversation discusses using the time-energy uncertainty relation to solve a problem involving the Schrodinger equation and a transcendental equation. The result is a formula for the time uncertainty, but there is some confusion about the physical dimensions of the result. The conversation ends with a suggestion to check the result using dimensional analysis.
Wannabe Physicist
Homework Statement
A particle moves in a 1D symmetric infinite square well of length ##2a##. Inside the well, the potential is ##V(x) = \gamma \delta(x)##. For sufficiently large ##\gamma##, calculate the time required for the particle to tunnel from being in the ground state of the well extending from ##x = -a## to ##x = 0## to the ground state of the well extending from ##x=0## to x=a##.
Relevant Equations
##\Delta t \Delta E \geq \hbar/2##
I am guessing time-energy uncertainty relation is the way to solve this. I solved the Schrodinger equation for both the regions and used to continuity at ##x=-a, 0,a## and got ##\psi(-a<x<0) = A\sin(\kappa(x+a))## and ##\psi(0<x<a) = -A\sin(\kappa(x-a))## where ##\kappa^2 = 2mE/\hbar^2##. Looking at the discontuity in the derivative, I got the transcendental equation
$$\tan(\kappa a) = -\frac{\hbar^2 \kappa}{m\gamma}$$
Setting ##x = \kappa a##,
$$\tan(x) = -\frac{\hbar^2 }{m\gamma a}x$$.

Imitating what one does for finite square well case, I plotted both the curves. One root is at ##x=0##. But that would mean ##\kappa = E = 0##. So that cannot represent a bound state. Next root for ##\gamma a = m/\hbar^2## (which means ##\tan(x) = -x##) is ##x=2.029##. This implies
$$\kappa = \frac{2.029}{a} = 2.029\frac{\hbar^2\gamma}{m}$$
This gives
$$E = \frac{2m\kappa^2}{\hbar^2} = \frac{2m}{\hbar^2}\left(\frac{4.117 \hbar^4\gamma^2}{m^2}\right) = \frac{8.234\hbar^2\gamma^2}{m}$$.

Now ##\Delta t \Delta E \geq \hbar/2##. But $$E = p^2/2m \implies \Delta E = p\Delta p/m= p\hbar/(2m \Delta x) = p\hbar/(2m a) = \sqrt{2mE} \hbar/(2ma) = \sqrt{\frac{2E}{ma^2}}\frac{\hbar}{2}$$

Thus $$\Delta E = \sqrt{\frac{2E}{ma^2}}\frac{\hbar}{2}$$

Finally,
$$\Delta t = \frac{\hbar}{2\Delta E} = \sqrt{\frac{ma^2}{2E}}$$

Substituting ##E = \frac{8.234\hbar^2\gamma^2}{m}## I get
$$\Delta t = 0.4964\frac{ma\gamma}{\hbar}$$

Does all of this make sense? Is there a simpler method and I am over-complicating an extremely simple problem?

I observe physical dimension of your result is
$$dim[\frac{ma\gamma}{\hbar}]=ML^2T^{-1}$$
, not T which we expect. Instead I see
$$dim[\frac{a\hbar}{\gamma}]=T$$

Last edited:
Yes. Thank you.

berkeman

## 1. What is the Time-Energy Uncertainty Relation?

The Time-Energy Uncertainty Relation is a fundamental principle in quantum mechanics that states that the more precisely we know the energy of a quantum system, the less precisely we can know its time and vice versa. This means that there is a limit to the precision with which we can measure both time and energy simultaneously.

## 2. How does this relation impact the Schrodinger Equation?

The Time-Energy Uncertainty Relation has a direct impact on the Schrodinger Equation, which is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. The relation introduces a trade-off between the precision of energy and time measurements, which is reflected in the mathematical form of the Schrodinger Equation.

## 3. Can the Time-Energy Uncertainty Relation be used to solve the Schrodinger Equation?

Yes, the Time-Energy Uncertainty Relation can be used to solve the Schrodinger Equation. By taking into account the uncertainty in time and energy measurements, we can obtain more accurate solutions to the Schrodinger Equation. This is particularly useful in systems with high energy or short time scales, where the uncertainty relation becomes more significant.

## 4. How does exploring the Time-Energy Uncertainty Relation help us understand quantum systems?

Exploring the Time-Energy Uncertainty Relation allows us to gain a deeper understanding of the behavior of quantum systems. By considering the trade-off between time and energy measurements, we can better understand the limitations of our observations and the inherent uncertainty in quantum systems. This helps us to develop more accurate models and predictions for these systems.

## 5. Are there any practical applications of using the Time-Energy Uncertainty Relation to solve the Schrodinger Equation?

Yes, there are practical applications of using the Time-Energy Uncertainty Relation to solve the Schrodinger Equation. For example, it has been used in the development of quantum technologies such as quantum computing and quantum sensing. It has also been applied in fields such as quantum chemistry and quantum optics to better understand and predict the behavior of quantum systems.

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