# Energy levels generally invariant under fixed V, changing T

• I
• bananabandana
In summary, the conversation discusses how fixing the volume and increasing the temperature affects the occupancy of energy levels in a given system. It is explained that the energy levels themselves do not change, but the boundary conditions for the solution of the TISE remain fixed, resulting in the same form of solution and density of states. It is also mentioned that the energy levels do not directly depend on temperature, but rather on the mean velocity of particles in thermodynamics.

#### bananabandana

Why is is true that for a given system, if I fix the volume and increase the temperature, you'd expect the occupancy of the energy levels to change, but not the levels themselves?

Can I think of this in terms of the fact that the boundary conditions for the solution of the TISE are fixed, such that we have the same form of solution? (and same density of states) Or is there some other explanation?

Thanks

Can I think of this in terms of the fact that the boundary conditions for the solution of the TISE are fixed, such that we have the same form of solution?
Yes.

The energy levels depend on whatever goes into the SE, if temperature is not there, then the energy levels do not depend on temperature.
In thermodynamics, temperature is the mean velocity of the particles... this would be related to the energy levels how?

## 1. What does it mean for energy levels to be invariant under fixed V?

When energy levels are invariant under fixed V, it means that the energy of a system remains constant regardless of any changes in volume. This is known as the law of conservation of energy, which states that energy cannot be created or destroyed, only transferred or transformed.

## 2. How does changing temperature affect energy levels?

Changing the temperature of a system can cause the energy levels to shift. As temperature increases, the average kinetic energy of the particles in the system also increases, resulting in higher energy levels. Conversely, decreasing the temperature can cause the energy levels to decrease.

## 3. Why are energy levels generally invariant under fixed V, changing T?

This is because the change in temperature does not affect the overall energy of the system, as long as the volume remains constant. The law of conservation of energy holds true, and any changes in temperature only result in a redistribution of energy among the particles.

## 4. Are there any exceptions to the rule of invariant energy levels under fixed V, changing T?

Yes, there are some exceptions to this rule. For example, in certain phase transitions, such as melting or boiling, there can be a change in energy levels even when the volume remains constant. This is because these transitions involve a change in the arrangement or movement of particles, which can affect the overall energy of the system.

## 5. How does this concept apply to real-life systems?

The concept of energy levels being invariant under fixed V, changing T applies to many real-life systems, such as a gas in a container or a chemical reaction in a closed system. In these cases, the energy of the system remains constant, even if the temperature changes, as long as the volume remains the same.