Solving the Time-dependent Schrödinger Equation for the 1D Particle in a Box

This video is part of my Quantum Mechanics for Physical Chemistry playlist:    • Quantum Mechanics   0:00 – Intro: What we’re solving today 0:14 – Why most lessons stop at the time-independent case 0:34 – Quantum dots as real-world “particles in boxes” 1:16 – How box size determines quantized energy levels 2:03 – What “one-dimensional particle in a box” means 2:38 – Infinite potential barriers and confinement 3:22 – Visualizing the 1D box with the marble-in-a-tube analogy 4:14 – Energy inside the box vs. infinity outside 4:33 – What is a complex conjugate? Real and imaginary parts 6:04 – Multiplying conjugates gives real numbers 6:23 – What is separability? Breaking multi-variable functions apart 8:12 – Introducing the full time-dependent Schrodinger equation 9:13 – Understanding it as a differential equation 11:16 – What wavefunctions represent and how they give probabilities 12:56 – Combining Schrodinger’s equation with separability 13:43 – Splitting position and time into separate equations 17:08 – Setting both sides equal to a constant (energy) 19:48 – Solving the time-independent, position-dependent part first 21:08 – Recognizing sine and cosine as natural solutions 24:00 – Applying boundary conditions inside the box 25:34 – Quantization emerges: only certain values of k are allowed 26:57 – Connecting wavefunctions to musical harmonics 28:42 – Deriving the quantized energy levels (energy proportional to n² over L²) 30:09 – Normalizing the wavefunction so total probability equals one 31:49 – Solving the normalization integral 33:29 – Finding the normalization constant (square root of 2 over L) 34:22 – Writing the full spatial wavefunction 34:47 – Solving the time-dependent part 35:18 – The surprising property that 1 divided by i equals negative i 36:03 – Checking the time-dependent exponential solution 38:19 – Combining space and time for the full wavefunction 39:33 – Visualizing the real and imaginary parts 41:24 – Oscillating amplitudes and phase relationships 42:23 – Multiplying by the complex conjugate to get probability density 43:33 – Stationary states: why probabilities don’t change with time 47:02 – Using the PhET simulator to visualize the wavefunction 48:21 – Identifying quantum states (n = 2, the first excited state) 48:39 – Superposition states and time-dependent probabilities 50:59 – Summary and invitation to explore with the PhET simulator