Relation between energy levels and volume.

[weezy]

I've just started with statistical mechanics and arrived at the part where they relate entropy to the number of microstates for a given system. The derivation starts of by adding an amount of heat $\delta Q$ to a system and observing the resulting change in internal energy : $$\delta U = \delta Q - P\delta V$$
where internal energy $U$ is given by $$ U = \sum _i N_iE_i $$ where $N_i$ is the no. of particles with energy $E_i$. Also: $$\delta U = \sum _i \delta N_iE_i + \sum _i N_i\delta E_i $$ Now it's stated here that
>the change $\delta E_i$ in energy level is only possible if volume changes

and hence: $$\sum _i N_i\delta E_i = \sum _i N_i\frac{\partial E_i}{\partial V}\delta V = -P\delta V $$ where $P=-\frac{\partial U}{\partial V}$

Using all equations we can see $$\delta Q = \sum _i E_i\delta N_i $$

I'm having trouble understanding why is change in energy level $\delta E_i$ only possible with change in volume. Any type of energy adding to the system should cause a change in energy level but what does it have to do with volume?

Also I'm not understanding what change in energy level means here actually. To me a particle can have any energy $E_i$ and when some energy is added to the system and let's assume that particle happens to absorb some of the energy it gets promoted to a higher energy level $E_i+\epsilon$ and but the previous energy level $E_i$ still exists. Only now $N_i - 1$ particles occupy energy $E_i$. So what exactly has changed here?

 

[Henryk]

Weezy,

In general, solutions to the Schrödinger equation (and energy level values) depend on the volume. The term $\delta E_i$ refers to the value of energy for a given state. Adding energy to the system does not change the energy levels, instead, it moves particles from lower energy levels to the higher ones. Only changing the volume can change values of energy levels.

Take, for example, an ideal gas. In quantum mechanics, the kinetic energy levels are quantized. The allowed levels are given by the 'particle in a box' solution (1-D example)
$$ E_n = \frac {\pi^2 \hbar ^2 n^2}{2mL^2}$$
This example gives you an explicit dependence of value of energy of a given state on dimension. Notice that the energy is well defined whether you have a particle in a given state or not.
You can differentiate with respect to L to get
$$\frac {dE_n}{dL} = 2\frac {\pi^2 \hbar ^2 n^2}{2mL^3} = \frac 2 L E_n$$ or
$$ \frac {dE_n}{dV} =\frac {dE_n}{AdL} = \frac 2 V E_n$$
From this, you can easily derive ideal gas law.
Henryk

 

[weezy]

Weezy,

In general, solutions to the Schrödinger equation (and energy level values) depend on the volume. The term $\delta E_i$ refers to the value of energy for a given state. Adding energy to the system does not change the energy levels, instead, it moves particles from lower energy levels to the higher ones. Only changing the volume can change values of energy levels.

Take, for example, an ideal gas. In quantum mechanics, the kinetic energy levels are quantized. The allowed levels are given by the 'particle in a box' solution (1-D example)
$$ E_n = \frac {\pi^2 \hbar ^2 n^2}{2mL^2}$$
This example gives you an explicit dependence of value of energy of a given state on dimension. Notice that the energy is well defined whether you have a particle in a given state or not.
You can differentiate with respect to L to get
$$\frac {dE_n}{dL} = 2\frac {\pi^2 \hbar ^2 n^2}{2mL^3} = \frac 2 L E_n$$ or
$$ \frac {dE_n}{dV} =\frac {dE_n}{AdL} = \frac 2 V E_n$$
From this, you can easily derive ideal gas law.
Henryk

Thank you so much for this answer. Much appreciated .

 

[weezy]

Just one more thing does this volume dependence of energy levels extend to other kind of systems which are not comprised of gases but rather solid matter? Say a ferromagnetic piece inside a magnetic field. What would govern the change in energy levels in such a scenario ?

 

[Henryk]

Statistical physics applies to all systems containing a large number of particles and that includes solids as well. An ideal gas is an easy example. In the case of solids, the electron energies do depend on distance between atoms but the calculations are much more difficult. But the effect of interatomic distances on the energy levels gives you elastic properties of solids such as Young modulus, etc.

Effects of external magnetic field is another story. Basically, the energy of a magnetic dipole in the presence of an external field is $E_m = - \vec \mu \cdot \vec B$ and this has to be added to the internal energy of the system (similarly for an electric dipole in the presence of electric field). So, yes, magnetic field acting on a ferromagnet does change energy but the change is not dependent on the volume and does not contribute to pressure. However, there is an effect called magnetostriction but that's the result of changing the size of magnetic domain.

Henryk

 

[weezy]

Statistical physics applies to all systems containing a large number of particles and that includes solids as well. An ideal gas is an easy example. In the case of solids, the electron energies do depend on distance between atoms but the calculations are much more difficult. But the effect of interatomic distances on the energy levels gives you elastic properties of solids such as Young modulus, etc.

Effects of external magnetic field is another story. Basically, the energy of a magnetic dipole in the presence of an external field is $E_m = - \vec \mu \cdot \vec B$ and this has to be added to the internal energy of the system (similarly for an electric dipole in the presence of electric field). So, yes, magnetic field acting on a ferromagnet does change energy but the change is not dependent on the volume and does not contribute to pressure. However, there is an effect called magnetostriction but that's the result of changing the size of magnetic domain.

Henryk

So if I were to compress a magnet such as to squeeze the interatomic distances a little closer I'm changing the boundary conditions and that is what causes changes in energy levels in solids?

 

[Henryk]

In solids you also have to include the electrostatic energy between the positive nuclei and negative electrons. Basically, an atoms other than hydrogen will have many levels occupied. The inner levels do not change when atoms form solids. It is the valence electrons whose energy is change by formation of a solid and their energy is dependent on the inter-atomic distances.

In solid state theory there are two simples models to compute energy levels of valence electrons. One is a Nearly Free Electron model. You start with electrons in a box. The boundary conditions traditionally used in solid state theory are somewhat different: they are so-called periodic von Karman that gives you a basis functions as plane waves. The kinetic energy term for this function will scale the same way with the size of the solid as for the 'particle in the box' solution, that is increases as you decrease the box size. Then you have to consider potential energy term. The potential energy has a mean component and periodic component because atoms are arranged in a regular way - they form a crystal.

For simple metals, like alkali, the periodic component is small and you have the mean component which actually decreases as you squeeze the atoms together. The equilibrium position is when the change of potential energy with distance balances out the change of kinetic energy change with distance. However, neither component depends on shape, just volume and these metals are very soft.

For most metals, the periodic component is not small and needs to be included, the calculations are a bit more complex. The net result is that energy depends on distance but also on the structural arrangement of atoms

The second simple model is called Linear Combination of Atomic Orbitals. The electron wavefunctions are taken as linear combination of atomic orbitals. When the atoms are brought together, the atomic orbitals of neighbouring atoms overlap and start interacting by an electrostatic repulsion between electrons. The net result is that the initial atomic orbitals which have equal energy (if the atoms are identical) are split into two new states, one of lower energy than the original (bonding) and another of higher energy (antibonding). As you squeeze the solid, you change the amount of the overlap between the neighbouring atoms and that changes the net interaction - hence energy.

The second approach is more applicable to ferromagnets. Ferromagnetis is a result of overlap between orbitals of neighbouring atoms with lower energy state having the same spin (or orbital momentum) orientation.

In short, in solids, the change of energy with dimension has kinetic energy component but potential energy component as well.
I hope I managed to explain more than confuse.

Henryk