A particle is confined in a potential well such that its...

A particle is confined in a potential well such that its allowed energies are $E_{n}=n^{2} \mathcal{E}$, where $n=1,2, \ldots$ is an integer and $\mathcal{E}$ a positive constant. The corresponding energy eigenstates are $|1\rangle,|2\rangle, \ldots,|n\rangle, \ldots$ At $t=0$ the particle is in the state $$ |\psi(0)\rangle=0.2|1\rangle+0.3|2\rangle+0.4|3\rangle+0.843|4\rangle $$ a. What is the probability, if the energy is measured at $t=0$, of finding a number smaller than $6 \mathcal{E} ?$ b. What is the mean value and what is the rms deviation of the energy of the particle in the state $|\psi(0)\rangle ?$ c. Calculate the state vector $|\psi\rangle$ at time $t$. Do the results found in (a) and (b) for time $t$ remain valid for arbitrary time $t ?$ d. When the energy is measured it turns out to be $16 \mathcal{E}$. After the measurement, what is the state of the system? What result is obtained if the energy is measured again?

A particle is confined in a potential well such that its allowed energies are $E_{n}=n^{2} \mathcal{E}$, where $n=1,2, \ldots$ is an integer and $\mathcal{E}$ a positive constant. The corresponding energy eigenstates are $|1\rangle,|2\rangle, \ldots,|n\rangle, \ldots$ At $t=0$ the particle is in the state
$$
|\psi(0)\rangle=0.2|1\rangle+0.3|2\rangle+0.4|3\rangle+0.843|4\rangle
$$
a. What is the probability, if the energy is measured at $t=0$, of finding a number smaller than $6 \mathcal{E} ?$
b. What is the mean value and what is the rms deviation of the energy of the particle in the state $|\psi(0)\rangle ?$
c. Calculate the state vector $|\psi\rangle$ at time $t$. Do the results found in (a) and (b) for time $t$ remain valid for arbitrary time $t ?$
d. When the energy is measured it turns out to be $16 \mathcal{E}$. After the measurement, what is the state of the system? What result is obtained if the energy is measured again?

Key Concepts

Wavefunction Collapse

Wavefunction collapse refers to the post-measurement change in the state of a quantum system. When an observable is measured, the quantum state instantaneously reduces to the eigenstate associated with the observed eigenvalue. This collapse eliminates the original superposition, and subsequent measurements of the same observable will yield the same result if the system remains isolated.

Time Evolution of Quantum States

The time evolution of a quantum state is governed by the Schrödinger equation. For systems with time-independent Hamiltonians, each eigenstate evolves by acquiring a phase factor dependent on its eigenvalue and time. Despite these phase changes, the probabilities associated with measurement outcomes of observables remain constant over time, provided the observable commutes with the Hamiltonian.

Expectation Value and Variance

The expectation value, or mean value, in quantum mechanics represents the average value of an observable, such as energy, one would obtain from a large number of measurements on identically prepared systems. The variance or rms deviation measures the spread or uncertainty in these measurements. These statistical measures are calculated using the state's amplitudes and the corresponding eigenvalues of the observable.

Measurement Probability and the Born Rule

The Born rule is a fundamental postulate of quantum mechanics that relates the probability of finding a particle in a particular state upon measurement to the squared absolute value of the amplitude associated with that state. When an observable like energy is measured, the probabilities calculated using the Born rule tell us how likely it is to obtain each of the eigenvalues of the observable.

Energy Quantization and Eigenstates

In quantum mechanics, when a particle is confined within a potential well, its energy becomes quantized, meaning that only specific, discrete energy values (eigenvalues) are allowed. The corresponding eigenfunctions or eigenstates provide the probability distributions for the location of the particle. The quantization stems from imposing boundary conditions on the wavefunction, and each energy eigenstate is associated with a unique eigenvalue.

Quantum Potential Well

A quantum potential well is a region in space where a particle is confined by a potential energy that is lower inside the region than outside it. This confinement leads to discrete energy levels, which are derived from solving the Schrödinger equation under boundary conditions appropriate to the well. The concept is central to understanding systems like electrons in semiconductors and other quantized states in confined geometries.

Quantum Superposition

Quantum superposition is the principle that a quantum system can exist in multiple states simultaneously, represented as a linear combination of its basis states. For example, a state vector can be expressed as a sum of different eigenstates with specific coefficients. The probability of measuring a specific eigenstate is determined by the square of the magnitude of its coefficient in the superposition.

Transcript

00:01 For this question we are given a particle in a potential square well and within the square well there is also additional potential that is described by the second equation over here now we're given that this is the wave function that describes the particle and we want to find what is the energy so energy can be described using the shuringer's equation so what we want to do over here in order to find e, we want to substitute in what is our side as well as our potential into this expression, and then hopefully get a term that is a constant multiplied by the wave function and this constant must correspond to the energy.

01:04 So let us start by finding out what is our second derivative of the wave function.

01:13 So we differentiate our wave function once, differentiate again.

01:48 Now we can put everything into the left -hand side of the shorninger's equation.

02:11 So we add the potential over here, multiplied by the wave function, which is a times 1 minus x squared over l squared.

02:33 Now if you are quick to notice, you should be able to identify that the bottom denominator for this particular term we can actually divide or factor out l square.

02:50 If you factor out of l square, we will get l to the power 4 outside.

02:55 But inside, you'll become 1 minus x square over l square which is the same term as the numerator which we can then divide away right so after simplifying this is what we have left now there should be quite a familiar term which you can all the common factors which you can just bring out so the common factors are a h bar square m and l square.

04:04 What we have left remaining is 1 minus x squared over l square.

04:12 And this should look very familiar to you.

04:17 This is just hbar square over m l square times the wave function side.

04:27 And this is exactly what we needed...

Please give Ace some feedback