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Due to comments below this answer I note the term "degenerate states" was subjective so firstly I'll give a formal definition:

Degenerate states results when two (or more) different eigenstates to an eigenvalue equation correspond to the same eigenvalue.

I'm tasked with the following question on a problem sheet set by my lecturer:

Do degenerate states always have the same energy?

The answer given was:

False (they have the same eigenvalue of any operator, not necessarily energy)

My interpretation of the quotation above is as follows: The eigenvalues of any operator can be the same but the energy can be different. At the moment it is my understanding that Eigenvalues=Energy Eigenvalues=Energies. Is there a distinction between these terms when considering the Hamiltonian? If my interpretation of the answer is wrong could someone please explain what the correct interpretation is? If you are able to do this there is no need to read any more of the content below until you reach the line, where below is my main question.

I do not understand how this answer can be false, so I will use the eigenvalues to the two dimensional Simple Harmonic Oscillator as an example that I believe contradicts the answer statement:

The TISE eigenvalue equation for the SHO is $$\hat Hu(x,y)=\left[-\frac{\hbar^2}{2m}\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+\frac{m\,{\omega_{x}}^2x^2}{2}+\frac{m\,{\omega_{y}}^2y^2}{2}\right]u(x,y)=E\,u(x,y)\tag{1}$$

Using separation of variables $u(x,y)=X(x)Y(y)$ such that $(1)$ becomes

$$-\frac{\hbar^2}{2m}\left(Y\frac{\partial^2X}{\partial x^2}+X\frac{\partial^2Y}{\partial y^2}\right)+\frac{m\,{\omega_{x}}^2x^2}{2}XY+\frac{m\,{\omega_{y}}^2y^2}{2}XY=EXY$$

and dividing throughout by $XY$ gives

$$\left(-\frac{\hbar^2}{2m}\frac{1}{X}\frac{\partial^2X}{\partial x^2}+\frac{m\,{\omega_{x}}^2x^2}{2}\right)+\left(-\frac{\hbar^2}{2m}\frac{1}{Y}\frac{\partial^2Y}{\partial y^2}+\frac{m\,{\omega_{y}}^2y^2}{2}\right)=E$$

Where $E=E_x+E_y$, I can now write equations involving $x$ and $y$:

$$-\frac{\hbar^2}{2m}\frac{\partial^2X}{\partial x^2}+\frac{m\,{\omega_{x}}^2x^2}{2}X=E_xX\tag{a}$$

$$-\frac{\hbar^2}{2m}\frac{\partial^2Y}{\partial y^2}+\frac{m\,{\omega_{y}}^2y^2}{2}Y=E_yY\tag{b}$$

Since $(\mathrm{a})$ and $(\mathrm{b})$ are one dimensional SHO eigenstate equations in $x$ and $y$ respectively; then we immediately know the eigenvalues:

$$E_x=\left(n_x + \frac12\right)\hbar\,\omega_x\quad\text{&}\quad E_y=\left(n_y + \frac12\right)\hbar\,\omega_y$$

Hence, the total energy $E$ is

$$E=E_x+E_y=\left(n_x + \frac12\right)\hbar\,\omega_x+\left(n_y + \frac12\right)\hbar\,\omega_y\tag{2}$$

So we now need two quantum numbers $n_x$ and $n_y$ to label eigenstates $$u_{{n_{x}}{n_{y}}}\equiv u_{n_{x}}(x)\,u_{n_{y}}(y)\tag{c}$$

Now suppose we are given a Central Potential such that $\omega_{x}=\omega_{y}=\omega_0$

Then $$V(x,y)=\frac{m\,{\omega_{x}}^2x^2}{2}+\frac{m\,{\omega_{y}}^2y^2}{2}=\frac{m\,{\omega_0}^2(x^2+y^2)}{2}=\frac{m\,{\omega_0}^2r^2}{2}=V(r)\tag{3}$$

As a result of $(2)$ and $(3)$ the energies are now given by

$$E=E_x+E_y=(n_x+n_y+1)\hbar\,\omega_0$$

The ground state clearly has $n_x=n_y=0$, for which $E_0=\hbar\,\omega_0$. However, there are now two first excited states, given by $n_x=1,n_y=0$ and $n_x=0,n_y=1$, both of which have $\fbox{$\color{blue}{E_1=2\hbar\,\omega_0}$}$. So from the definiton $(\mathrm{c})$ I have two degenerate states: $u_{10}$ and $u_{01}$ with eigenvalue equations $$\hat H u_{10}=E_1 u_{10}$$ and $$\hat H u_{01}=E_1 u_{01}$$

So I have found a case where same eigenvalues $E_1$ are the same energy.

Is this just a one-off coincidence, or can anyone provide me with an example where the energies are different but the Eigenvalues are the Same?


More importantly suppose we have a free particle with $$E=\frac{{\hbar}^2 k^2}{2m}$$

and a wavefunction $\psi$ satisfying $$\frac{d^2 \psi}{dx^2}+k^2\psi=0\tag{4}$$

$(4)$ has two possible solutions: $$\psi_1(x)=Ae^{ikx}\quad\text{and}\quad\psi_2(x)=Ae^{-ikx}$$

Applying the momentum operator $$\hat p=-i\hbar\frac{d}{dx}$$ to the two wavefunctions that solve $(4)$ gives for $\psi_1(x)$ $$\hat p\psi_1(x)=-i\hbar\frac{d}{dx}\left(Ae^{ikx}\right)=-i\hbar(ik)Ae^{ikx}=(\hbar k)Ae^{ikx}=\color{red}{(\hbar k)}\psi_1(x)$$ Similarly for $\psi_2(x)$ $$\hat p\psi_2(x)=\color{red}{-(\hbar k)}\psi_2(x)$$

So I have two degenerate states: $\psi_1(x)$ and $\psi_2(x)$ that apparently share eigenvalues.

But $$\color{red}{\hbar k} \ne \color{red}{-\hbar k}$$ If the eigenvalues are different how can the states be degenerate?

Or put in another way; since $\psi_1(x)$ and $\psi_2(x)$ no longer have the same eigenvalue this violates the definition of degenerate states in the first quotation at the beginning of this post.


EDIT:

It was my understanding that the minus sign belongs to the eigenvalue.

Do we simply consider the absolute value of the eigenvalue so that $\color{red}{\hbar k}$ will be the same for $\psi_1(x)$ and $\psi_2(x)$?

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    $\begingroup$ What does the book mean by degenerate states? $\endgroup$ – Valter Moretti May 27 '17 at 16:32
  • $\begingroup$ As @ValterMoretti implies I'd say that it depends on your definition. Usually, you state that some solutions are degenerate if they have the same eigenvalue for the Hamiltonian (i.e. they have the same energy). But it would seem that your lecturer uses degeneracy in a more general context so that degenerate solutions may have the same eigenvalues for whatever operator. Note that this would require the lecturer to state: these solutions are degenerate with respect to operator $\hat{X}$. $\endgroup$ – gertian May 27 '17 at 17:07
  • $\begingroup$ In your example the modes would be degenerate with respect to $\hat{H}$ but not with respect to $\hat{H_x}$ $\endgroup$ – gertian May 27 '17 at 17:08
  • $\begingroup$ Actually I cannot understand well the question in any sense. In general one speaks about degenerate levels rather than degenerate states, meaning eigenvalues (energies) of the Hamiltonian operator whose eigenspaces have dimension greater than $1$. $\endgroup$ – Valter Moretti May 27 '17 at 18:29
  • $\begingroup$ So, in general I may agree with your initial answer...however the point is that "degenerate states" is not standard terminology so, without a precise definition, it is not possible to answer. $\endgroup$ – Valter Moretti May 27 '17 at 18:32
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Question: do degenerate states always have the same energy?

The answer depends on what one means by degenerate. This word typically means two or more states with the same energy, in which case the answer is trivially yes.

On the other hand, it seems that according to your professor, degenerate means two or more states that share the eigenvalue with respect to some operator. This definition is not particularly useful, because all states are eigenvectors of the identity, with eigenvalue one, which means that all states are degenerate. And even if we restrict ourselves to operators other than the identity, the notion of degenerate is still trivial, because one may consider arbitrary projectors of the form $$P=|\varphi_1\rangle\langle\varphi_1|+|\varphi_2\rangle\langle\varphi_2|$$ which would make $\varphi_1,\varphi_2$ degenerate, for any pair of states $\varphi_1,\varphi_2$. In this sense, this definition of degenerate is vacuous.

Finally, if by degenerate we mean eigenvectors that share the eigenvalue with respect to a particular operator $A$ (fixed, i.e., not arbitrary), then the question becomes more interesting: given a pair of states $a_1,a_2$, such that $$ \begin{aligned} A|a_1\rangle&=a|a_1\rangle\\ A|a_2\rangle&=a|a_2\rangle \end{aligned} $$ then do we necessarily have $$ \begin{aligned} H|a_1\rangle&=E|a_1\rangle\\ H|a_2\rangle&=E|a_2\rangle \end{aligned} $$ for some energy $E$?

The answer is not necessarily, as can be seen by constructing simple counter-examples.

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  • $\begingroup$ Hi, thanks for your answer, please see my edited version as there is another question at the bottom of the post that has two eigenvalues that are supposedly equal even though they differ by a minus sign. Any ideas on that one? Regards. $\endgroup$ – user138066 May 29 '17 at 23:05
  • $\begingroup$ @user395550 I'm not sure I understand your edit. You found two states with the same energy but different momentum. What's bugging you about that? $\endgroup$ – AccidentalFourierTransform May 31 '17 at 21:48
  • $\begingroup$ What's bugging me is that this violates the defintion of degeneracy (in the quote at the beginning of the post) as the two states $\psi_1(x)$ and $\psi_2(x)$ are not both equal to $\color{red}{\hbar k}$. They no longer share eigenvalues. $\endgroup$ – user138066 Jun 3 '17 at 4:55
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    $\begingroup$ @user395550 please note that "degenerate" does not mean that the eigenvalues are the same for all operators. It just means that they are the same for at least one operator. In this case, the energy eigenvalue is the same, even though the momentum eigenvalue is not. In other words, $E_1=E_2$ and $p_1\neq p_2$. As the energies are the same, the states are degenerate. $\endgroup$ – AccidentalFourierTransform Jun 3 '17 at 14:29
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    $\begingroup$ @user395550 because $E=\frac{{\hbar}^2 k^2}{2m}$, as you wrote in the OP. In this case, $k_1=-k_2$, and therefore $k_1^2=k_2^2$ and therefore $E_1=E_2$. $\endgroup$ – AccidentalFourierTransform Jun 3 '17 at 18:04
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The most common reference to the word 'degenerate' is when we refer to the energy eigenstates. If the eigenvalue of the energy operator is same for more than one state of the system, such states are called degenerate. \begin{aligned} H|a_1\rangle&=E|a_1\rangle\\ H|a_2\rangle&=E|a_2\rangle \end{aligned}

There is also a different definition of degenerate, in a general sense. If two or more eigenstates for some fixed("given") operator say A have the same eigenvalue, then those states are called as degenerate. The interesting observation about eigenstates is that if you have two eigenstates say, \begin{aligned} |a_1\rangle, |a_2\rangle \end{aligned} for some observable A, then any linear combination of them is also an eigenstate and hence, we have a 2 dimensional eigenspace, and we have a degree of freedom in the choice of the two eigenvectors from the eigenspace. Similarly, the more the degeneracy, the higher the dimension of the eigenspace.

Okay, now let's suppose you want to uniquely specify a quantum state. What do you do? This is how we can do it :-

1) You find the eigenvalues corresponding to some observable,(say X) but it turned out to show degenerate values for some of it's eigenstates. Hence, just labeling the states using only X is not enough to distinguish the states uniquely.

2)You find another observable which commutes with X,(say Y) so that it has the same set of eigenvectors as X. The eigenvalues of this operator provided unique values for some of the eigenvectors and hence, a greater number of states can now be uniquely specified but suppose there are still eigenvectors which are degenerate that is have the same eigenvalue for both X and Y. What do we do? We execute step 3.

3)Find another such commuting observable and keep on doing so till every eigenstate can be labeled uniquely. This is why we say that in QM, the state of any system is specified by the Complete set of Commuting Observable's.

The above mentioned thought process was carried out to show the role of degeneracy and what it exactly is, in a general sense. There is much more to degenracy than just this. For example, Symmetry of a system with respect to some observable results in degenrate states of that observable.

As I am a 2nd year undergrad. Hence, some of the things mentioned might be wrong. If anyone wants to correct or add something, feel free to do so. I love critics. They help us be better versions of ourselves!!

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