What is the Schrodinger equation in one dimension?

What is the Schrodinger equation in three dimensions?

What is the probability density?

In one dimension, the probability is proportional to ψ^{2}(x)Δx, where ψ^{2}(x) is the probability density. If Δx is small, the probability is ψ^{2}(x)dx. In three dimensions, Ψ^{2}(x,y,z)ΔxΔyΔz=ψ^{2}(x,y,z)ΔV. If ΔV is very small, Ψ2(x,y,z)dxdydz=ψ2(x,y,z)dV.

How do you normalise the wavefunction?

If ψ is a solution to the Schrodinger equation, so is Nψ, where N is any constant. It becomes the normalisation constant.

What are acceptable wavefunctions?

They must be single-valued at any point. They must not be infinite over a measurable interval. They must be a continuous and smooth function. Their first derivative must be a continuous and smooth function. They must be quadratically integrable so the integral must be finite.

What is the Heisengberg uncertainty principle?

It is not possible to specifiy simultaneously, with arbitrary precision, both the momentum and the position of the particle.

What is the Schrodinger equation for a particle in a one-dimensional box?

How do you find the solution for the Schrodinger equation for a particle in a one-dimensional box?

What is the effect of the boundary conditions on the Schrodinger equation for a particle in a one-dimensional box?

How do you normalise the wavefunction to calculate the constant A?

What is the energy equation for a particle in a one-dimensional box?

Describe a quantum particle in a one-dimensional box.

The kinetic energy is quantised. The lowest energy level (known as the zero-point energy) is h^{2}/(8mL^{2}). This agrees with the Heisenberg uncertainty principle. For n=1, the maximum probability density ψ^{2} is at x = L/2; for n=2 the maxima are at x = L/4 and x = 3L/4. A point at which ψ=ψ^{2}=0 is called a node. Generally, the number of nodes increases with the increase in energy. The energy n^{2}h^{2}/(8mL^{2}) decreases with the increase of the mass of the particle m and the size of the box L. The effects of quantization become less important, the energy gradually becomes continuous and the particle begins to behave as expected in classical mechanics.

Describe a classical particle in a one-dimensoinal box.

The kinetic energy varies continuously and can take any value. The lowest value that the energy can take is zero. This corresponds to a stationary particle. The probability of finding a moving particle anywhere in the box would be same irrespective of the energy. No differences for large m and/or large L.

Describe the free electron model.

One approximate way of treating the π electrons in the double bonds of a polyene is to assume that they move along the conjugated chain as particles in a one-dimensional box. The potential energy inside the box (along the chain) is constant but rises sharply at the ends.

What is tunnelling?

It is observed when the barrier is not infinite.

How do you calculate the probability of a particle tunnelling through the barrier?

What is the Schrodinger equation for a particle in a three-dimensional box?

What is the solution for the Schrodinger equation of a particle in a three-dimensional box?

What is the energy equation for a particle in a three-dimensional box?

If the three-dimensional box is a cube, what is the wavefunction and energy of the particle?

What is the Schrodinger equation for a particle in a ring?

Give the probability density, normalisation integral and overlap integral of real and complex wavefunctions.

What is the energy equation for a particle in a ring?

Apply the boundary conditions to the wavefunction of a particle in a ring and normalise it.

The boundary condition is that after a complete turn around the ring, the wavefunction should remain unchanged.

What is the equation for angular momentum?

m = mass

v = velocity

r = radius

p = linear momentum

How do you calculate angular momentum from kinetic energy and total energy?

What is the probability density of a particle in a ring?

The probability density is independent of the angle φ. The momentum and angular momentum of a particle in a ring have exact values, but the position is completely undefined.

What is Hook's law?

What is the equation for the potential energy of a compressed or extended spring?

What are the allowed energies of the quantum harmonic oscillator?