process of predicting future evolution values

  1. take current state
  2. decompose it into a superposition of the energy eigen-states
  3. evolve complex exponential factors
  4. add the results


time evolution calculation

  • time evolution of particle in a box
  • time evolution in the harmonic oscillator

superposition for a particle in a box

  • consider infinitely deep potential well
    • this is the ‘particle in a box’ problem
    • with a particle in it
  • consider the particle to be in a linear superposition
    • with equal parts of the first and second states of the well

\[ \begin{align} \psi(z,t) = \frac{1}{\sqrt{L_z}} \left[ \exp{\left( i \frac{E_1}{\hbar}t \right)} \sin{\left( \frac{\pi z}{L_z} \right)} + \exp{\left( i \frac{E_2}{\hbar}t \right)} \sin{\left( \frac{2\pi z}{L_z} \right)} \right] \tag{1} \end{align} \]

  • wave function for first eigen state of well: \[ \sin \left( \frac{\pi z}{L_z} \right)
    \]

  • corresponding complex exponential time-varying function for first eigen state of well: \[ \exp{ \left( i \frac{E_1}{\hbar} t \right) } \]

  • wave function for second eigen state of the well: \[ \sin \left( \frac{2\pi z}{L_z} \right) \]

  • corresponding complex exponential time-varying function for second eigen state of the well: \[ \exp{ \left( i \frac{E_2}{\hbar} t \right) } \]

  • normalization factor of the superposition: \[ \frac{1}{\sqrt{L_z}} \]
    • this is the normalization for the superposition
    • the individual normalizations simplify to this superposition
    • this is the same for eigen-states (1 and 2 superposition) and (2 and 3 superposition)
  • the probability density is obtained by multiplying (1) by its complex conjugate \[ \lvert \Psi (z,t) \rvert^2 = \frac{1}{L_z} \left[ \sin^2\left( \frac{\pi z}{L_z} \right) + \sin^2 \left( \frac{2\pi z}{L_z} \right) + 2 \cos \left( \frac{E_2 - E_1}{h} t \right) \sin\left( \frac{\pi z}{L_z} \right) \sin{ \left( \frac{2\pi z}{L_z} \right) } \right] \tag{2} \]
    • the complex conjugate is obtained by flipping the sign of \(i\) everywhere it exists \[ \omega_{21} = \frac{E_2 - E_1}{\hbar} = \frac{3E_1}{\hbar} \]
  • absolute energy origin does not matter here for this measurable quantity
    • only energy difference \( E_2 - E_1 \) matters

eigen functions of a particle in a box

  • first few particle-in-a-box energy levels
    • with their associated wave-functions

particle-in-a-box

  • orange dashed lines are horizontal axes


superposition

\( n = 1\)

  • consider \(n = 1\) spatial eigen-function \( \psi_1(z) \) n=1

  • probability density of wavefunction @ \(n = 1\) and \( \lvert \psi_1(z) \rvert^2 \) n=1

  • the eigen energy wave function is sinusoid
    • it’s probability density is gaussian

  • multiplying by the time dependent factor gives \[ \Psi_1(z,t) = \exp{ \left( -i \frac{E_1}{\hbar} t \right) } \psi_1(z) \]

  • the probability density functions for both TD and TI wave functions are the same \[ \lvert \Psi_1(z,t) \rvert^2 = \lvert \psi_1(z) \rvert^2 \]

  • multiplying the TI with the exponential to make it TD compatible
    • does not change the probability density function of the resulting TD wave function
    • nothing measurable changes in the process of converting it from TI to TD


\( n = 2 \)
  • wavefunction

    • this is an odd function

    • wavefunction


  • probability density

    • this is a positive function

    • probability-density


  • multiplying by the time dependent factor gives

\[ \Psi_2(z,t) = \exp{ \left( -i \frac{E_2}{\hbar} t \right) } \psi_2(z) \]

  • the probability density functions for both TD and TI wave functions are the same

\[ \lvert \Psi_2(z,t) \rvert^2 = \lvert \psi_2(z) \rvert^2 \]

  • multiplying the TI with the exponential to make it TD compatible
    • does not change the probability density function of the resulting TD wave function
    • nothing measurable changes in the process of converting it from TI to TD


resulting superposition wave function

  • merging QMly to obtain superposition involves
    1. adding the two wave function amplitudes and
    2. taking the squared modulus to get the probability density
    3. computing the superposition oscillation from the difference between the angular frequencies of the two waves
      • assuming it is equal superposition

super-position-wave

  • individually, the probability density of the first two TD eigen-functions are stationary
    • if they are simply mathematically added, the result is stationary
      • with no oscillation


  • an equal superposition of the two oscillates at the angular frequency \[ \begin{align} \omega_{21} & = \frac{E_2 - E_1}{\hbar} \\
    & = \frac{3E_1}{\hbar} \\
    \end{align} \]

\[ \begin{align} \lvert \Psi(z,t) \rvert^2 & = \lvert \Psi_1(z,t) + \Psi_2(z,t) \rvert^2 \\
& = \lvert \psi_1(z) \rvert^2 + \lvert \psi_2(z) \rvert^2 + 2\cos{ \left( \frac{E_2 - E_1}{\hbar} t \right)} \psi_1(z)\psi_2(z) \\
\end{align} \]

  • this is taken to imply particle in well is oscillating from left to right within the particle well
    • when the 1st and the 2nd eigen-functions’ probability density functions are merged


superposition for the harmonic oscillator

  • similar to particle in a box example
    • the first two eigen energy states are explored
  • consider two energy eigen-states in a simple harmonic oscillator
    • \( E_a \)
    • \( E_b \)

superposition oscillation frequency

  • superposition wavefunction is given by \[ \Psi_{ab}(\textbf{r},t) = c_a \exp{ \left( -i \frac{E_a}{\hbar} t \right)} \psi_a(\textbf{r}) + c_b \exp{ \left( -i \frac{E_b}{\hbar} t \right) }\psi_b(\textbf{r}) \]

  • probability distribution of this superposition wavefunction is \[ \begin{align} \lvert \Psi_{ab} (\textbf{r},t) \rvert^2 & = \lvert c_a \rvert^2 \lvert \psi_a(\textbf{r}) \rvert^2 + \lvert c_b \rvert^2 \lvert \psi_b(\textbf{r})\rvert^2 + 2 \lvert c_a^* \psi_a^{*}(\textbf{r}) \rvert \cos{\left[ \frac{(E_a - E_b)t}{\hbar} - \theta_{ab} \right]} \\
    \text{ where: } \\
    \theta_{ab} & = arg( c_a \psi_a(\textbf{r}) c_b^{*}\psi_b^{*}(\textbf{r})) \\
    \end{align} \]

eigen functions and probability densities

  • consider two harmonic energy levels (plotted in orange) in a QM harmonic oscillator
    • with associated wave-functions
    • energy-levels


\( n = 0 \)

  • consider the TI spatial eigenfunction \( \psi_0(z) \)
    • for \( n = 0 \)
    • wavefunction


  • probability density of above eigenfunction \( \psi_0(z) \) is given by

    • \( \lvert \psi_0(z) \rvert^2 \)

    • probability-density
    • probability density of superposition oscillates at the angular frequency

\[ \omega_{ab} = \frac{\lvert E_a - E_b \rvert}{\hbar} \]

  • multiplying TI wavefunction \( \psi_0(z) \) by the TD-factor gives \[ \Psi_0(z,t) = \exp{ \left(-i \frac{E_0}{\hbar} t \right) }\psi_0(z) \]
    • multiplication does not change the probability density of the \[ \lvert \psi_0(z,t) \rvert^2 = \lvert \psi_0 \rvert^2 \]


\( n = 1 \)

  • consider the spatial eigenfunction of \( \psi_1(z) \)
    • for \( n = 1 \)
    • Imgur


  • probability density function of \( \psi_1(z) \)

    • Imgur


  • multiplying by the exponential time-dependent (TD) factor \( \exp{ \left( -i \frac{E_1}{h} t \right) } \)

\[ \begin{align} \Psi_{1}(z,t) = \exp{ \left( -i \frac{E_1}{\hbar} t \right) } \psi_1(z) \end{align} \]

  • the probability densities are the same \[ \begin{align} \lvert \Psi_{1}(z,t) \rvert^2 = \lvert \psi_{1}(z,t) \rvert^2 \end{align} \]


  • an equal superposition of the two oscillates at the angular frequency \[ \begin{align} \omega & = \frac{(E_1 - E_0)}{\hbar} \\
    \lvert \Psi(z,t) \rvert^2 & = \lvert \Psi_0{(z,t)} + \Psi_1{(z,t)} \rvert^2 \\
    \end{align} \]

  • Oscillation of Equal Superposition

  • simplifying expression of equal superposition \[ \begin{align} \lvert \Psi(z,t) \rvert^2 & = \lvert \Psi_0{(z,t)} + \Psi_1{(z,t)} \rvert^2 \\\ \lvert \Psi(z,t) \rvert^2 & = \lvert \Psi_0{(z,t)} \rvert^2 + \lvert \Psi_1{(z,t)} \rvert^2 + 2\cos{(\omega t)} \psi_0(z) \psi_1(z) \\
    \end{align} \]

the coherent state

  • the QM harmonic oscillator above does not correspond well with the classical oscillator
    • QM does not show a single well defined probability distribution moving smoothly from one side to another
    • though the behavior of the resulting probability distribution oscillates from one side to another
      • it had various bumps in it
      • changes shape as it oscillates
  • the above superposition is a simple one
    • with only first two levels of the simple harmonic oscillator
  • many other different superpositions can be created


  • there exists one state of superposition called the coherent state
    • it is a linear superposition
    • as the overall energy of this superposition is increased, progressively, the behavior of a classical harmonic oscillator is recovered
    • forms part of the modern quantum mechanical theory of lasers and coherent light


  • the coherent state for a harmonic oscillator of frequency \( \omega \) is \[ \Psi_N(\xi,t) = \sum_{n=0}^{\infty} c_{Nn} \exp{ \left[ -i \left( n + \frac{1}{2} \right) \omega t \right] } \psi_n(\xi) \tag{1} \]
    • where: \[ c_{Nn} = \sqrt{\frac{N^n \exp{(-N)}}{n!}} \]
    • the \( \psi_n(\xi) \) are the harmonic oscillator eigen-states


  • the note that for the expansion coefficients \( C_{Nn} \) \[ \lvert c_{Nn} \rvert^2 = \frac{N^n \exp{(-N)}}{n!} \]
    • this is poisson distribution from statistics
    • with mean \(N\) and standard deviation \( \sqrt{N} \)
    • the arrival of the number of photons in a laser beam follows this poisson distribution
      • a laser is in its ideal form in a coherent state
  • the justification for why classical harmonic oscillators are best represented by coherent states
    • is beyond the scope of this lecture notes
  • \( N\) in (1) can have any real value
    • it represents a statistical average of some kind
    • not a quantum number